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in this video we're going to cover a few common Concepts that you'll see in a typical algebra course so let's get right into it so let's say if you have two terms 5x + 4x these are considered like terms whenever you have like terms you're allowed to add the coefficients 5 + 4 is 9 so 5x + 4x is 9x now let's see if you have another expression 3x + 4 y + 5x + 8 y you can't add 5x and 8 y because they're not like terms however you can add 3x and 5x that's going to equal 8X and you can
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also add 4 Y and 8 y that's going going to be 12 y so you can add the coefficients of like terms another example let's say if you have 3 radical 2 + 5 radical 7 + 8 radical 2 + 3 radical 7 these are like terms because the radical 2 is the same you can add the three three and five not three and five but the 3 and 8 3 + 8 is 11 so it's 11 < tk2 now these two are like terms you can add the five and the three to give you eight so it's 8un 7 try this one 7 x + 4x^ 2 + 5x + 9 x^2
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so we can add the 4x2 and a 9x2 because they're like terms they both have an x squ 4 + 9 is 13 x^2 we can add the 7 x and a 5x 7 and five is 12 so this is the
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answer try this one let's say if you have
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9 x^2 + 6 x + 5 plus 3x^2 - 5x - 9 so notice that we can add 9x2 and 3x2 they're like terms so that's going to be 12 x^2 now we can also add 6 x and 5X X 6x +5x is the same as 6x - 5x 6 - 5 is 1 so we're just going to get 1 x 5 +9 5 - 9 is-4 so the final answer is 12 x^2 + x -
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4 here's another example you can try 3x 2 + 7 x - 4 - 8 x^2 - 5x + 7 now notice that we have a negative sign we need to distribute the negative sign to the three terms on the right so for the three numbers on the left we don't have anything in front of the parentheses so we could simply open it we don't need the parentheses on the left side so it's simply 3x^2 + 7x - 4 now you can treat this as a negative 1 so let's distribute the negative sign to everything on the right so instead of
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having positive 8X s it's going to be negative 8x^2 and instead of having - 5x it's going to be a positive 5x and instead of having positive s if we distribute the negative sign to it it's 7 and so now let's add like terms 3x^2 +8 x^2 3 - 8 is5 so it's - 5x^2 7 x + 5x 7 + 5 is 12 so it's 12x -4 + -7 is the same as4 - 7 which is1 and so that's how you can add or subtract
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polinomial so what is a polom a polom is a function with many terms a monomial is simply one term 8X is a monomial 5x^2 that's a monomial or 3 x^2 Y is a monomial a binomial has two terms like 5x + 6 X 7X - 3 those are binomials a trinomial has three terms so x^2 + 6 x + 5 is a trinomial a polom is an expression that has many terms now let's say if we wish to multiply a monomial by a trinomial
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what is the answer what do we need to do so we need to distribute 7x to everything inside so what's 7 x *
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x^2 so x^2 is the same as 1 x^2 7 * 1 is 7 now what's x * x^2 x is the same as x to the first Power * x^2 whenever you multiply variables you need to add the exponents 1 + 2 is three so this is X Cub x to the first power is simply X x^2 is x * X together you have a total of 3 X's multiply to each other that's why it's X
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Cub therefore 7 x * x^2 is 7 x Cub now what is 7 x * 2x 7 * 2 is 14 x * X or X the 1 power * X the 1 power 1 + 1 is 2 so it's X2 and then 7 x * -3 is - 21x so that's how you can multiply a monomial by trinomial so now it's your turn try this 5x^2 time 3x to 4th power - 6 x Cub + 5x - 8 so let's distribute the 5x^2 5x^2 * 3x 4 5 * 3 is 15 x^2 * x 4 is x 6 you need to add 2 and 4 2 + 4 is 6 so now we need to multiply 5 x^2 * -6x Cub 5 * -6 is -30 x^2 * X Cub 2 + 3 is 5
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so it's going to going be x to the 5th power and then 5x^2 * 5x 5 * 5 is 25 X2 * X we have 2 + 1 which is 3 and so it's simply going to be X cub and then 5 x^2 * 8 that's 5 * 8 is4 so it's -40 x^2 So This Is The Answer um to this problem now let's multiply a binomial by another binomial so what's 3x - 4 * 2x + 7 so what we need to do here is something called foil we need to foil these two expressions what's 3x * 2x 3 * 2 is 6 and x * X is x^2 keep in mind this is X to the first power 1 + 1 is
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2 so next we need to multiply the 3x by the 7 3x * 7 is simply 21x -4 * 2x is -8x and -4 * 7 that's going to be -28 so now we need to combine like terms the like terms that we have in this expression is 21x and 8x so what's 21 - 8 21 - 8 is pos3 or POS 13x so this is the final answer that's how you can foil two binomial Expressions so the answer is 6x^2 + 13x - 28 try this example 2x - 5 and 4x + 7 go ahead and foro this expression so 2x * 4X that's 8 x^2 and 2x * 7 is 14x -5 and 4x is
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-2X and5 * 7 is -35 so let's add like terms 14x - 20x is -6x so this is the final answer to this particular problem now what if you see an expression that looks like
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this what's 2x - 3^ 2 how would you simplify this expression so 2x - 3^ 2 is the same as 2x - 3 * 2x - 3 so you have to foral again 2x * 2x is 4x^2 2x * -3 that's -6x -3 * 2x is also -6x and finally -3 * -3 two negatives will change into a positive now just like before we need to combine like terms 6 + 6 is 12 so -6 + -6 is -12 So This Is The Answer 4x^2 - 12x + 9 now let's say if we wish to multiply a binomial which has two terms with a trinomial which has three terms
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so initially after we multiply we should get a total of 2 * 3 terms which is six terms so let's multiply the 5x by 2x^2 5 * 2 is 10 x * x^2 is X Cub because 1 + 2 is 3 so now let's multiply 5x * -3x 5 * -3 is15 x * X is x^2 next let's multiply 5x * 4 5x * 4 is 20x and -9 * 2x^2 that's 8x^2 9 * -3x that's positive
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27x and we have one more -9 * pos4 that's -36 so now let's add like terms and let's arrange it so we have 10 x Cub we can combine -15 and8 that's -33 x^2 and we combine these two 20x + 27x that's 47x and there's nothing to combine the 36 with so we're just going to leave it as 36 so this is the answer to this particular
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problem now let's review some properties of exponents what's X Cub * x 4 power whenever you multiply common variables you need to add the X exponents 3 + 4 is 7 so this is x 7 now what is X to the 9 / x 4 whenever you're dividing you need to subtract the exponents 9 - 4 is five now what about when you raise one exponent to another exponent what do you need to do you need to multiply this is 7 * 6 which is 42 now let's explain why these rules work the way they
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do so let's start with multiplication we said that x^2 * X Cub is X 5th power keep in mind x^2 is simply x * x x Cub is x * x * X there's three X's multiplied so we have a total of 5 X's multiplied that's why when you multiply common variables you need to add the exponents as you can see we have 5 x's in this expression now why is it that when we raise one exponent to another we need to multiply instead of add it's 2 * 3 not 2 + 3 why is it x 6 x^2 to the thir power means that we have 3 X2
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and x^2 each is x * X so we have a total of six x variables multiply to each other that's why you need to multiply one exponent by the other whenever you raise an exponent by another
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one now what about when dividing let's say if we have x to the 5ifth power / x^2 we said it's going to be 5 - 2 which is 3 so this is X Cub another way in which you can see it is that X to 5th power is equal to x * x * x * x * X it's 5 X's multiply to each other X2 is simply x * X so you can cancel 2 X's on top and on the bottom and you're left over with three X's which is therefore x to the 3 so it's the same as doing 5 minus 2 which is three so now what about this example let's say if we have
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X to 4th / x to the 7th what is the answer so 4 - 7 top minus bottom is -3 and whenever you have a negative exponent if you move it to the bottom that is if you move the variable to the bottom bottom the negative exponent will become positive so x^ -3 is the same as 1X cub and typically in algebra you want your final answer to contain positive exponents not negative exponents so this is the answer but now let's see if we can understand it X to 4th we know it's x * x * x * x x 7 is x * x * x 7
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* so we can get rid of four x variables therefore we have three left over on the bottom and there's no more on top so 3 X's together is X cub and thus we could see why this is the answer so now let's work on some practice problems what is 3 x to 4th y 5th multiplied 5 x to 6 Y 7th so first we need to multiply the three and the five 3 * 5 is 15 next we need to multiply x 4 * x 6 that's going to be 4 + 6 which is 10 and then we need to multiply y 5th * y 7 5 + 7 is 12 so the answer is 15 x 10 y 12
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try this example what is 8 x Cub y -2 ultip 7 x to 8 y 5 so first we need to multiply 8 and 7 8 * 7 is 56 x 3r * X8 3 +8 is5 so it's X to5 and then y --2 * y 5 -2 + 5 is 3 so we have y Cub notice that we have a negative exponent and we need to make it positive therefore we need to move X from the top to the bottom so the answer is going to be 56 YB / x to the 5th
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now what if we have a problem that looks like this 24 x 7 y -2 / 6 x 4 y 5 what's the answer so we need to divide let's divide 24 by 6 24 ID 6 is 4 now what's x 7 / x 4 we need to subtract top minus bottom 7 7 - 4 is 3 after you subtract it initially it goes on top now what about the next one y^ -2 ID y the 5th power so -2 - 5 to subtract it you can use the number line so here's zero here's -2 so you need to go five units to the left whenever you subtract move to the left
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of the number line if you need to add move to the right so 1 2 3 4 five this is -3 -45 -67 so -2 - 5 is7 and initially it goes on top so now to make the negative exponent positive we need to move the Y to the bottom so the final answer is 4X Cub / Y 7th try this one 32 x to the 5th power y^ -3 Z to 4th / 40 x to8 y^7 z to the8 so we can't divide 30 by I mean 32 by 40 nicely however notice that 8 can go into 32 and 40 so let's divide the top number and and the bottom number by 8 32 / 8 is 4
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40 / 8 is 5 so now we could focus on the variables x to the 5th power / X to8 so 5 -8 5 -8 is the same as 5 + 8 which is 13 so initially it's going to go on top so we have X to 13 on top now what y -3 / y7 so what's -3 - -7 -3 - -7 is the same as -3 + 7 which is 4 so we have y^ the 4th power now for the Z's it's going to be 4 minus8 which is the same as 4 + 8 so that's 12 so this is the answer we don't have any negative exponents so we can leave the answer like this
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try this one 3 x Cub raised to the second power what is it equal so keep in mind whenever you raise one exponent to another you need to multiply so 1 * 2 is 2 and 3 * 2 that's 6 so we have 3^2 x 6 3^ 2 is the same as 3 * 3 3 * 3 is 9 so the final answer is 9 x 6 by the way what is the difference between 5x^2 and 5 + x^2 are these the same 5x^2 is basically 20 5 x^2 but 5 + x^2 is not 5^2 + x^2 it doesn't work that way instead you need to foil this is 5 + x * 5 + x which we've covered already
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so notice that the five and X are multiplied therefore you can distribute the two to the five and the X but here we have a plus sign in between whenever you see that you need to expand it like this and foil so 5 + x * 5 + x is going to be different 5 * 5 is 25 5 * X that's 5x x * 5 is also 5x and finally x * X is x^2 so this is equal to 25 + 10 x + X2
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let's try another example try this one what is four x^2 y 3 raised to the 3 power so four is the same as four to the first power so let's distribute the three 3 * 1 is 3 3 * 2 is 6 and 3 * 3 is 9 so what's 4 to the 3 power 4 the 3 power is 4 * 4 * 4 4 * 4 is 16 and 16 * 4 is 64 so this is the answer to that expression now what if you get a question that looks like this let's say it's 8 x^2 y 5th Z 6 raised to the 0 power anything raised to the zero power is always one it doesn't matter what's
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inside now let's say if you have -2 5x y Cub raised to Z what's the answer now so everything inside the parentheses is going to turn into a one so only this portion will become a one but notice that you have a -2 on the outside -2 * 1 is -2 so that's the answer for an expression that looks like that
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now let's say if you get an expression that looks like this 5 x -2 / y -3 * 8 x 4 / y5 what's the answer to this particular problem now the first thing I would do is try to get rid of the negative exponents so x to the -2 we can move it to the bottom it's going to become x^2 and the Y we can move it to the top and the same is true for the y^ the neg 5 if we move it to the top it's going to be y to the positive uh five so now let's multiply across 8 * 5 is 40 and Y Cub time y 5th we need to add
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the exponents it's going to be y to the 8 since 3 + 5 is 8 and for the X variables we need to divide X to 4 / X2 that's 4 - 2 so it's simply 2 so the final answer is 40 x^2 y
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8 let's try this one 35 x to -3 / 40 x y the 5th power * 24 x^2 y^2 / 42 y4 so how can we simplify this expression what do you think we need to do well what you don't want to do is you don't want to multiply across if you multiply 35 and 24 you're going to get a large number and if you multiply 40 and 42 you're going to get a bigger number and then you have to reduce the fraction it's a lot of work instead we need to simplify or break down these numbers into smaller numbers we can break down
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35 into 7 and 5 and 40 we can make that 8 * 5 now the X to -3 I'm going to move it to the bottom so I have x * X the 3r on the bottom and Y 5th 24 I'm going to rewrite that as 8 * 3 and 42 is 7 * 6 but six I'm going to write that as 3 * 2 now y^ the4 I'm going to move it to the top to make it y to the POS 4 so now let's see what we can cancel we can cancel an8 we can get rid of a three and we can get rid of a seven and a five
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so on top we have an x
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s y^ 2 * y 4 2 + 4 is 6 so it's y 6 on top on the bottom we have a two we have x * X3 which is X to the 4th and we also have y to the 5th so now 2 - 4 is -2 but if we subtract it backwards 4 Min - 2 is 2 and since the number on the bottom is bigger we're going to have X2 left over on the bottom if you do 2 minus 4 you're going to get -2 on top and then you'll have to move it to the bottom which will make it positive two for the Y's we have 6 - 5 which is 1 and that's going to stay on top so this
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is the final answer for this particular problem so now it's your turn try this one 24x y / 27 x^ -2 / by 36 x^2 Y3 over 45 XY to 4th now how can we divide these two expressions there's something called Keep Change Flip you need to keep the first fraction the same way and then you could change division to multiplication if you flip the second fraction so now we can solve it the same way as we did in the last
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example so let's rewrite 24 as 8 * 3 27 we can make it 9 * 3 and we can move the X to the -2 to the top 45 let's make it 9 * 5 and 36 we could make that 9 * 4 or 6 * 6 so let's change 24 instead of making it 8 * 3 let's make it 6 * 4 and 36 is going to be 6 * 6 and so we have x^2 y to3 I'm going to move that to the
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top so now let's cancel a six we can cancel a nine and that's about it well actually the four we can change that into 2 * 2 and we have a five we have three on the bottom six we can make that 3 * 2 and that was this six that remained so now let's multiply the variables that we have on top so we can also cancel an x s by the way so we left over with x x * X on top which is x^2 and we also have y * y 4 * y 3r so 1 + 3 + 4 is 8 so we have y 8 on top so we can cancel a two so what we
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have left over is 2 * 5 which is 10 X 2 y to 8 and on the bottom we have 3 * 3 which is 9 so this is the final answer now consider the expression 3x over 5 ided by 7x y over 9 how can you simplify this expression so what we have here is a complex fraction we have a fraction within a fraction you can rewrite this as 3x 5 / 7x y 9 that's what it really means and so using Keep Change Flip we can keep the first fraction the same change division to multiplication and flip the second
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fraction so now we can figure out the answer so we can cancel X and then we have to multiply across 3 * 9 is 27 5 * 7 is 35 So This Is The Answer 27 over 35 y now let's say if you have a complex fraction that looks like this 7 + 2x / 5 - 3 over Y how would you simplify this
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expression the best way to simplify is to multiply top and bottom by the common denominator of these two fractions the common denominator is X and Y so let's multiply the top part by X x y and the bottom part by XY so XY * 7 is simply 7 XY x y * 2x the X will cancel and so what we have left over is 2 * y now for the bottom part x * y * 5 is 5X Y and then XY * 3 y y cancels so so it's 3x but -- 3x now if we want to we can factor a y if you take out a y on top it's going
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to be 7 x + 2 and on the bottom you can factor out an X if you take that out it's going to be 5 Yus 3 and so that's the answer so now we're going to focus on solving equations so let's say say if we have the equation x + 4 is = to 9 how would you solve for x x is simply a variable it's an unknown number you want to find out what number plus 4 is 9 you know intuitively that 5 + 4 is 9 so X is equal to 5 but let's solve it to solve for x you need to get X on one side of the equation which
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means that you need to move the four to the other side the opposite of addition is subtraction so you need to subtract both sides by four to solve for x four and4 cancels and so 9 - 4 is 5 and that's how you can solve for x let's try another example what is 3x + 5 = 11 what is the value of x the first thing we need to do is subtract five from both sides
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11 - 5 is 6 so now how can we separate the three from the X notice that the three is Multiplied to the X the opposite of multiplication is division so we need to divide both sides by three 6 / 3 is 2 so that is the value of x and to check it let's plug in two into the equation 3 * 2 + 5 we know that 3 * 2 is 6 6 + 5 is 11 and so two is the correct answer for X let's try this one 2 * x -1 + 6 is = to 10 so what's the first thing that we need to do to solve for
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x the first thing I would do is subtract six from both
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sides so 10 - 6 is equal to 4 now to get rid of the parentheses we can either distribute 2 to x - one or we could divide both sides by two and so on the left side since we no longer have a two in front of the parentheses we can get rid of the parentheses and so it's just x - one on the left side on the right side four divid 2 is 2 so to isolate X we just got to add one to both sides 2 + 1 is three so therefore X is equal to 3 try this one 5 - 3 * x + 4 = 7 + 2 * x
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-1 so in this particular example let's distribute the -3 and the 2 first so -3 * X is simply -3x and -3 * 4 is -12 now let's distribute the two on the right side 2 * X is 2X and 2 * -1 is -2 so now let's add like terms 5 and -12 5 - 12 is -7 on the right side we can combine 7 and -2 7 - 2 is 5 so now let's add 3x to both sides and simultaneously let's subtract five from both sides we need to do this in such a way that we can get X on one side of the equation so 2X and 3X they add to 5x -7 +5 is
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-12 so therefore 5x is equal to -12 to separate the five from the X we need to divide both sides by five 5 / 5 is 1 so X is equal to to -12 over 5 as an improper
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fraction here's another one what if we have 2 over 3x + 5 is equal to 8 how would you solve for x the first thing I would do is subtract both sides by five so therefore 2 over 3x is = to 8 - 5 which is 3 now what we could do is multiply both sides by three on the left side the threes will cancel so all you have on the left side is 2x on the right side 3 * 3 is 9 so to separate the two from the X we need to divide both sides by two so the final answer is X is equal to 9 over 2 which is about
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4.5 let's try this one 3 over 4 x - 1/3 is equal to 1 how can we solve for x in this particular example when you have many fractions it's going to help a lot if you attempt to clear away all the fractions let's multiply both sides of the equation by the common denominator the common denominator of 4 and 3 is 12 so let's distribute 12 to each term so what's 34 x * 12 you can do 3 * 12 / 4 or you can do 12 / 4 * 3 3 * 12 is 36 36 over 4 is 9 but if you do 12/ 4
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that's three times the three on top you still get 9 either way it's going to be 9x now what's 12 * 1/3 12 * A3 is the same as 12 ID 3 which is 4 and then 12 * 1 is 12 so let's add four to both sides 12 + 4 is 16 and if we divide both sides by 9 we can see that X is equal to 16 over
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9 try this one x + 2 over 5 is equal to 7/ 8 so how can we solve for x if we're given two fractions separated by an equal sign if you get a problem like this you can cross multiply 5 * 7 is 35 and 8 * x - 2 you need to distribute the 8 it's going to be 8 x + 16 so to solve for x let's subtract both sides by 16 so 35 - 16 let's subtract it 5 - 6 is a negative number so let's borrow a one so 15 - 6 is 9 2 - 1 is 1 so 35 - 16 is
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19 so what we now have is 8x is equal to 19 so let's divide both sides by 8 so X is 19 8 if you want to convert it to a mixed fraction need to realize that 8 goes into 19 two times 8 * 2 is 16 and 19 - 16 is three so it's 2 and 38 as a mixed number what if you have an equation that contains decimal numbers how would you solve for
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x notice that most of the numbers that we have here there's two digits after the decimal point so therefore we need to multiply both sides by 100 if there was only one digit after the decimal point we'd multiply both sides by 10 so we're got to multiply every number by 100 04x * 100 you simply need to move the decimal two units to the right so it's going to be 4x5 * 100 is 15 9x * 100 is 9x 25 * 100 is 25 so now let's subtract 4X from both sides and simultaneously let's add 25 to both
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sides so these will cancel 15 + 25 is 40 9x x - 4x is 5X so to isolate X we need to divide both sides by five so 40 ID 5 is 8 so that is the value of x for this particular problem and as you can see it's not that
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bad consider this equation how can we solve for x there's two ways in which we could solve for x the first thing we could do is we can add 25 to both both sides so x^2 is equal to 25 at this point we could take the square root of both sides the square root of x^2 is simply X and the square root of 25 is plus orus 5 so X can equal 5 or X can equal 5 5 * 5 is 255 * 5 is also 25 so that's one way in which you can solve an equation that looks like that another technique that you can use is
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you can Factor this expression using the difference of squares method the square root of x^2 is x the square root of 25 is five on one side you're going to have a positive sign and on the other side you're going to have a negative sign so factoring is the opposite of for L now you need to set each factor equal to zero so for the first one if you subtract both sides by five you'll see that X is equal to5 and for the second expression if you add both sides by five you'll see
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that X can also be equal to postive 5 try this one 2x^2 - 18 is equal 0 how can we solve for x X so we can't Square < TK 2 and 18 we won't get a nice number however we can factor out the GCF the greatest common factor we can take out a two from 2x^2 and 18 to find out what goes inside divide 2x^2 / 2 is X2 --8 ID 2 is -9 now notice that we can Factor x^2 - 9 using the difference of squares technique the square root of x^2 is x the square root of 9 is three and so it's going to be plus and minus
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therefore X can equal -3 and posi
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3 let's try this one 3 x^2 - 48 = 0 feel free to pause the video and try this example it's very similar to the last two so the first thing we need to do is we need to remove the GCF which is three three can go into 3x^2 and 48 3x^2 ID 3 is x^2 and -48 / 3 is -16 which we can factor using the difference of squares technique the square root of x s is x the square root of 16 is 4 and so it's going to be x + 4 and x - 4 so therefore X will equal -4 and
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positive4 now let's say if you have an expression that looks like this x 4 - 81 is equal to0 what is the value of x so notice that we can use the difference of squares technique the sare < TK of x 4 is X2 because x^2 * X2 is x 4 theun of 81 is nine so one side is going to be plus and the other side is going to be minus now we can't factor a sum of squares but we can Factor x^2 - 9 because that's still a difference of squares and that's going to be x + 3 * x -
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3 so therefore the real solutions that we have for x is -3 and
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posi3 by the way the factor x^2 + 9 can never be zero if you subtract both sides by 9 you'll see that x^2 is 9 which can't be if you plug in 3 3^2 is POS 9 if you plug in3 -3 * ne3 is postive 9 so you can't take the square root of a negative number the square root of 9 is not a real solution it's an imaginary solution this is equal to 3 I where I is equal to1 so you won't get a real answer for X if you want to look for an imaginary answer then it's equal to plus orus 3
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I now let's say if we have a trinomial x^2 - 5x + 6 isal to zero how can we Factor this expression to solve for x so notice that the leading coefficient is one what you need to do is find two numbers that multiply to six but that add to the middle term 5 so 1 and six won't work 2 and three is very close 2 * 3 is 6 but 2 + 3 is POS 5 but -2 and -3 still multiplies to positive 6 but add to5 so to factor it it's going to be x - 2 * x - 3 so if you set each factor equal to
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zero we could solve for x here we need to add two to both sides so we could see that X is equal to pos2 and for this one we need to add three to both sides so X is equal to
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pos3 try this one x^2 - 2x - 15 solve for x so what two numbers numbers multiply to -5 but add to the middle number -2 so it's not 1 and 15 3 and5 could work 3 * 5 is5 but 3 +5 is -2 so it's going to be x + 3 * x - 5 therefore X is equal to3 and posi 5
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try this one x^2 + 3x - 28 is equal to zero so what two numbers multiply to -28 but add to three so we have 2 and 14 and 4 and 7 4 and7 adds up tog3 but4 and positive 7 adds up to posi 3 but still multiplies to -28 so it's going to be x - 4 * x + 7 which means that X is equal to pos4 you need to change the sign and X isal to -7 so that's the solution to the equation now what if you have a trinomial where the leading coefficient is not one how can you factor this expression in order to solve for
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x in this case you need to multiply 2 and -2 2 * -2 is4 you need to find two numbers that multiply to4 but still add to the middle term 3 so this is going to be 4 and 1 4 +1 is pos3 but 4 * 1 is still4 now what we're going to do is we're going to replace the middle term 3x with 4X and --1x notice that the value of the expression is still the same 4X -1 X is 3x now our next step is to factor by grouping so let's Factor the GCF from the first two terms and the
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last two terms and the first two terms we can take out a 2X 2x^2 / 2x is X and 4x / 2x is 2 in the last two terms let's take out a 1 -1x / -1 is simply positive X and -2 / -1 well that's going to be pos2 so notice that we have a common factor X+ 2 when you see that that means you're on the right track if they're different you need to go back and check your work because it's an error somewhere so now we're going to factor out x + 2 so what's going to be on the inside of
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the second parenthesis is the stuff on the outside that's the 2X and the negative one so now at this point we can Factor the expression I mean not Factor but solve it let's set each factor equal to zero and let's solve for x so for this one all we need to do is subtract both sides by two and so X is equal to -2 now for the other one we got to do a little bit more work we need to add one to both sides so 2x is equal to 1 and then we need to divide both sides by two
54:27
so X is equal to 12 so X isal to -2 and X is = to a half now sometimes if you're having difficulty factoring an expression you can always use the quadratic equation so let's use the quadratic equation for the example that we just uh worked on now this equation is called a quadratic equation and it's in standard form that's ax plus I mean ax^2 + BX + C so therefore you could see that a is 2 B is three and C is -2 so using the quadratic equation it's B plus or minus the square < TK of b^2 -
55:20
4 a c / 2 a so it's going to be -3 because B is 3 plus or minus b^ 2 or 3^ 2 - 4 * a which is 2 * C which is -2 / 2 a or 2 *
55:46
2 so we have -3 plus or minus 3^ 2 is 9 -4 * 2 is8 and8 * 2 is 16 on the bottom 2 * 2 is 4 and 9 + 16 is 25 and the square root of 25 is five so right now we could separate this into two equations -3 + 5 / 4 and -3 - 5 over 4 because we have a plus minus -3 + 5 that's 2 so it's 2 over 4 and-3 - 5 is8 2 over 4 is the same as 12 which is the first answer that we have8 over 4 is -2 that's the second answer so if you're having difficulty solving it you can always use
56:43
the quadratic equation to get the answer let's try another problem like the last example 6 x^2 + 7 x - 3 is equal to 0 so solve this quadratic equation by factoring and by using the quadratic formula so the leading coefficient is not one so we need to multiply 6 and -3 6 * -3 is8 now what two numbers multiply to8 but add to the middle term 7 this is 9 and -2 9 * -2 is8 but 9 + -2 is 7 so we're going to replace 7x with 9x and -2X it doesn't matter the order in which you put the two numbers
57:36
you'll still get the same answer so now let's Factor by grouping in the first two terms take out the GCF 3x can go into 6x^2 and 9x so let's remove 3x 6x^2 / 3x is 2X and 9x / 3x is simply 3 in the last two terms let's take out a netive 1 -2X /1 is 2x -3 id1 is 3 so we have a common factor of 2x + 3 so let's take that out and within the other parenthesis it's going to be the 3x and the minus one so now let's set each factor equal to zero so 2x + 3 is = to 0 and 3x - 1 is equal
58:33
0 so here let's subtract both sides by three so 2x is equal to3 next let's divide both sides by two so therefore X is -3 over 2 now for the next example let's add one to both sides and so we can see that 3x is equal to 1 and then let's divide div both sides by three so X is 1/3 so these are the answers posi 1/3 and -3 over2 so now let's use the quadratic equation to get the same
59:18
answer so X is = to B plus orus < TK B - 4 a c/ 2 a so the first number in front of X2 this is a a is 6 b is 7 C is -3 so -7 plus or minus b^ 2 or 7^ 2 - 4 * 6 * -3 / 2 a or 2 *
59:51
6 7^ SAR is 49 and 6 * 4 is 24 24 * 3 20 * 3 is 60 4 * 3 is 12 60 and 12 is 72 so 24 and 3 is 72 and it's two negative signs so it's going to be positive 72 2 * 6 is 12 so now what's 49 + 72 so if we do it by hand 2 + 9 is 11 carry over the 1 4 and 1 is 5 + 7 that's 12 so we have 121 inside the
1:00:36
radical the square root of 121 is 11 so it's -7 plus or- 11 /
1:00:47
12 so we can separate it into two fractions -7 + 11 / 12 and -7 - 11/2 -7 + 11 that's 4 -7 - 11 is8 now 4 over 12 we can divide both numbers by four and so we can reduce it to 1 over three now for the next one we can divide both numbers by six 18 ID 6 is 3 12 divid 6 is 2 so we get the answers that we had in the last example so it's -3 over2 and 1/3 consider this expression X Cub - 4x^2 - x + 4 is equal to Z how would you solve for x notice that the first two terms has
1:01:41
the same ratio as the last two terms 1 and4 And1 and 4 whenever you see that and if you have a total of four terms you can Factor by grouping so in the first two terms let's take out an x s X Cub ID X2 is X -4x 2id X2 is4 and the last two terms let's take out of 1x /1 is X pos4 /1 is4 so we have a common factor of x - 4 now what we have left over on the outside is x^2 minus 1 which we can factor that further using the difference of squar method so it's going to be x +1 * x
1:02:30
-1 so the solutions are pos4 -1 and positive 1 so we have three answers for a cubic
1:02:44
function now the next topic that we're going to go over is graph and linear equations so how can we graph the equation y = 2x - 1 you need to realize is that this is in slope intercept form which is MX plus b so M represents the number in front of X so m is equal to 2 m represents the slope so the slope is two that's rise over run B is the Y intercept that's where it touches the y- axis B is Nega 1 so with this information we can graph the function
1:03:31
so the first thing we should do is plot the Y intercept which is NE -1 on the Y AIS so this is the Y AIS this is the x axis next we can use a slope to find the next point we said the slope is two or 2 over one which is rise over run so we need to rise two units up and then travel or run one unit to the right so therefore the next point is going to be at 1 comma 1 so then we're going to rise to run one so the next point is going to be over here at this point we can draw a
1:04:12
straight line and so that's how you can graph an equation in slope intercept form so let's try another example go ahead and graph y = 3 4 x - 2
1:04:45
so we can see that the Y intercept which is B is -2 so it's going to be on the Y AIS and the slope which is the number in front of x 34s so that means we need to travel or go up three units and then four units to the right the rise is three the run is four so the next point is going to be 4 comma one and so we could connect these two points with a straight line and that line wasn't straight so let's try that again and so that's how you can graph it now sometimes you might have an
1:05:24
equation in standard
1:05:29
form standard form is ax plus b y is equal to C now in this form if you wish to graph it the best thing to do is to find the X and Y intercept to find the x intercept substitute zero for y so this completely disappears so 2x is equal to 6 6 / 2 is 3 so X is so therefore the x intercept is 3 comma 0 next let's find the Y intercept to find the Y intercept substitute zero for X so therefore this disappears so -3 Y is equal to 6 6 / -3 is -2 so we have the Y intercept of 0
1:06:21
-2 now we can plot those two points and connect them with a straight line
1:06:39
so 30 is over here and 0 -2 is over here and then just connect them so that's how you can graph an equation in standard form now sometimes you may need to write the equation of the line given a point and a slope so there's three forms this form is the slope intercept form this is the standard form and this is the point slope form so what we're going to do for each of these examples maybe not each of them but some of them we're going to start with the point slope form converted to
1:07:25
the slope intercept form and then convert that to the standard form so you know how to find all three if ever the need arises so let's say if the slope is two and you have the point 1 comma 3 how would you write the equation of the line that passes through the point 1 comma 3 and that has a slope of two so personally I think it's easier if you start with the point slope equation so this is X1 and this is y1 so let's replace y1 with three M with two and X1 with one so this is the answer in
1:08:00
point slope form this is it it's y - 3 is equal to 2 * x -1 now if you want to convert it to the slope intercept form distribute the two 2 * X is 2X and 2 * -1 is -2 now let's add three to both sides the goal is to solve for y whenever you want it in point slope form not point slope form but slope incept form if you want it in slope intercept form isolate y so Y is equal to 2x + 1 so this is the answer in slope intercept form and this is the answer in point slope form
1:08:55
now to convert it to standard form we simply need to get X and Y on the left side of the equation so let's subtract both sides by 2x so therefore the equation in standard form is -2x + y is equal to 1 so now it's in ax + b yal c
1:09:17
form now how can we write the equation of the line if we're given two points let's say the point 2A 4 And1 comma 5 how can we write the equation of the line in this case we need to find a slope which is Y2 - y1 over X2 - X1 so this is going to be X1 y1 and this is X2 Y2 so it's going to be 5 - 4 / -1 - 2 5 - 4 is 1-1 - 2 is-3 so therefore the slope is- 13 so now let's write the equation of the line first in point slope form so y - y1 is = to M * x - X1 so y1 is 4 we can use either Point 2
1:10:13
4 or-15 I'm going to use 2 4 so y1 is 4 M is- 1/3 and X1 is 2 so this is the answer in point slope form so now let's convert converted to slope intercept form let's distribute the 1/3 it's going to be 13x and then- 1/3 * -2 -1 * -2 is 2 so it's POS
1:10:43
2/3 now we need to add four to both sides so it's going to be Y is = - 13x + 2 over 3 + 4 four is the same as 4 1 to add 2/3 and four we need to get common denominators so let's multiply 4 by 3 over
1:11:13
3 so 4 * 3 is 12 so 2/3 + 12/3 2 and 12 is 14 so this is the final answer in slope intercept form you can see that the slope is - 1/3 and the Y intercept is 14 over3 so now let's put this equation in standard form so to put it in standard form the first thing we need to do is get rid of the fractions so let's multiply both sides by three so 3 * Y is 3 y and 3 * - 13x the 3es cancel so it's simply --1x and 3 * 14 over 3 the 3es cancel so you just get 14 so now we need to move the X
1:12:13
to the other side so let's add X to both sides so the equation in standard form is going to be x + 3 Y is equal to 14 so now it's in ax + b y = C4 now sometimes you may need to write the equation of the line given another parallel or perpendicular line so let's say if you want to write the equation of the line that passes through the point 1 comma 3 and that's
1:12:51
parallel to the equation 2x - 3 y - 5 is equal to Z how would you do it keep in mind to write the equation of any line you need the point which we already have and a slope so we could find a slope using this equation since it's parallel the slopes will be the same so let's put that equation in slope intercept form and the number in front of X in that form will be the slope so let's solve for y so starting with this
1:13:30
equation let's subtract both sides by 2X and let's add five at the same time so on the left side we're going to have -3 Y and on the right side it's going to be -2x + 5 so now all we need to do is divide by3 to each ter so the threes are going to cancel on the left side so therefore Y is positive 2 over 3x - 5/3 so therefore we can see that the slope which is the number in front of X is 2/3 so now we can write the equation of the line let's write it in slope
1:14:17
intercept form so instead of using the point slope equation we're going to use this one directly so we're going to substitute y for 3 we're going to plug in 2/3 for M and replace x with one now let's solve for b so let's multiply both sides by three to get rid of the fraction so 3 * 3 is 9 and 3 * 2/3 the 3's cancel so you get two and don't forget to distribute the three to B so it's 3B so now let's subtract both sides by two so 9 - 2 is 7 7 is equal to 3B so
1:15:03
now we could divide both sides by three to isolate B so B is equal to 7 over 3 so now we can write the equation of the line in slope intercept form so y = mx plus b all we need to do is replace m and b so m is 2 over3 and b is 7 over3 so this is the equation of the line that is parallel to this line but passes through the 13 so here's the last question for the day write the equation of the line that passes through the point -2 comma 1 and that's perpendicular to the
1:15:46
equation 3x + 2 y - 7 is equal to0 so just like before first we need to convert this equation to standard form not standard form but slope intercept form once it's in slope intercept form we can find a slope and then we can find the slope of the perpendicular line so let's solve for y so let's move the 3x and the Seven to the other side let's subtract both sides by 3x and let's add seven to both sides so 2 Y is = to -3x + 7 and now let's divide each number by two so Y is = to
1:16:36
-3/2x + 7 /2 so the slope of this line is -3 over two therefore the slope of the perpendicular line it's going to be the negative reciprocal of that value so it's going to be positive 2 over3 you need to change the negative sign to a positive sign and flip the fraction at the same time so now that we have the point and the slope we can write the equation so let's use the slope intercept form equation so let's replace y with one M with 2/3 and x with -2 now to get rid of the fraction let's
1:17:16
multiply everything by three so 3 * 1 that's 3 3 * 2/3 * -2 the threes will cancel and so what we have left over is 2 * -2 which is just4 and then 3 * B which is 3B so now let's add four to both sides so therefore 7 is equal to 3B and now let's divide both sides by three so b is 7 over3 so now we can write the equation so let's plug in the values for m and b so it's going to be 2 over 3 x + 7 over3 so this is the equation of the line
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