SOLUTION: factor 80-130x+45x^2

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Question 432574: factor 80-130x+45x^2
Found 3 solutions by rwm, jorel1380, richard1234:
Answer by rwm(914) About Me  (Show Source):
You can put this solution on YOUR website!
5*(9x^2-26 x+16)
5 (x-2) (9x-8)
or
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


45%2Ax%5E2-130%2Ax%2B80 Start with the given expression.



5%289x%5E2-26x%2B16%29 Factor out the GCF 5.



Now let's try to factor the inner expression 9x%5E2-26x%2B16



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Looking at the expression 9x%5E2-26x%2B16, we can see that the first coefficient is 9, the second coefficient is -26, and the last term is 16.



Now multiply the first coefficient 9 by the last term 16 to get %289%29%2816%29=144.



Now the question is: what two whole numbers multiply to 144 (the previous product) and add to the second coefficient -26?



To find these two numbers, we need to list all of the factors of 144 (the previous product).



Factors of 144:

1,2,3,4,6,8,9,12,16,18,24,36,48,72,144

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to 144.

1*144 = 144
2*72 = 144
3*48 = 144
4*36 = 144
6*24 = 144
8*18 = 144
9*16 = 144
12*12 = 144
(-1)*(-144) = 144
(-2)*(-72) = 144
(-3)*(-48) = 144
(-4)*(-36) = 144
(-6)*(-24) = 144
(-8)*(-18) = 144
(-9)*(-16) = 144
(-12)*(-12) = 144


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -26:



First NumberSecond NumberSum
11441+144=145
2722+72=74
3483+48=51
4364+36=40
6246+24=30
8188+18=26
9169+16=25
121212+12=24
-1-144-1+(-144)=-145
-2-72-2+(-72)=-74
-3-48-3+(-48)=-51
-4-36-4+(-36)=-40
-6-24-6+(-24)=-30
-8-18-8+(-18)=-26
-9-16-9+(-16)=-25
-12-12-12+(-12)=-24




From the table, we can see that the two numbers -8 and -18 add to -26 (the middle coefficient).



So the two numbers -8 and -18 both multiply to 144 and add to -26



Now replace the middle term -26x with -8x-18x. Remember, -8 and -18 add to -26. So this shows us that -8x-18x=-26x.



9x%5E2%2Bhighlight%28-8x-18x%29%2B16 Replace the second term -26x with -8x-18x.



%289x%5E2-8x%29%2B%28-18x%2B16%29 Group the terms into two pairs.



x%289x-8%29%2B%28-18x%2B16%29 Factor out the GCF x from the first group.



x%289x-8%29-2%289x-8%29 Factor out 2 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



%28x-2%29%289x-8%29 Combine like terms. Or factor out the common term 9x-8



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So 5%289x%5E2-26x%2B16%29 then factors further to 5%28x-2%29%289x-8%29



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Answer:



So 45%2Ax%5E2-130%2Ax%2B80 completely factors to 5%28x-2%29%289x-8%29.



In other words, 45%2Ax%5E2-130%2Ax%2B80=5%28x-2%29%289x-8%29.



Note: you can check the answer by expanding 5%28x-2%29%289x-8%29 to get 45%2Ax%5E2-130%2Ax%2B80 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by jorel1380(3719) About Me  (Show Source):
You can put this solution on YOUR website!
80-130x+45x^2=45x^2-130x+80
"""""=9x^2-26x+16
"""""=(9x-8)(x-2)
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 9x%5E2%2B-26x%2B16+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-26%29%5E2-4%2A9%2A16=100.

Discriminant d=100 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--26%2B-sqrt%28+100+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-26%29%2Bsqrt%28+100+%29%29%2F2%5C9+=+2
x%5B2%5D+=+%28-%28-26%29-sqrt%28+100+%29%29%2F2%5C9+=+0.888888888888889

Quadratic expression 9x%5E2%2B-26x%2B16 can be factored:
9x%5E2%2B-26x%2B16+=+9%28x-2%29%2A%28x-0.888888888888889%29
Again, the answer is: 2, 0.888888888888889. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-26%2Ax%2B16+%29

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
It is somewhat tedious factoring such expressions with a relatively large leading coefficient, so we factor by finding the roots of the polynomial. If 9x%5E2+-+26x+%2B+16+=+0, then = 2 or 8/9. Hence our polynomial can be factored as C%28x-2%29%28x+-+8%2F9%29 where C is a constant. Since the leading coefficient (x%5E2 term) is 45, then C = 45, so the polynomial is equal to 45%28x-2%29%28x+-+8%2F9%29. For sake of neatness, we can distribute the 9 into the x+-+8%2F9 to obtain integer coefficients: 5%28x-2%29%289x-8%29.