SOLUTION: 20a squared + 39a + 8 factor out completely

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Question 656705: 20a squared + 39a + 8
factor out completely
Found 3 solutions by MathLover1, lynnlo, Alan3354:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=20+x%5E2%2B39+x%2B8 Start with the given equation



y-8=20+x%5E2%2B39+x Subtract 8 from both sides



y-8=20%28x%5E2%2B%2839%2F20%29x%29 Factor out the leading coefficient 20



Take half of the x coefficient 39%2F20 to get 39%2F40 (ie %281%2F2%29%2839%2F20%29=39%2F40).


Now square 39%2F40 to get 1521%2F1600 (ie %2839%2F40%29%5E2=%2839%2F40%29%2839%2F40%29=1521%2F1600)





y-8=20%28x%5E2%2B%2839%2F20%29x%2B1521%2F1600-1521%2F1600%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 1521%2F1600 does not change the equation




y-8=20%28%28x%2B39%2F40%29%5E2-1521%2F1600%29 Now factor x%5E2%2B%2839%2F20%29x%2B1521%2F1600 to get %28x%2B39%2F40%29%5E2



y-8=20%28x%2B39%2F40%29%5E2-20%281521%2F1600%29 Distribute



y-8=20%28x%2B39%2F40%29%5E2-1521%2F80 Multiply



y=20%28x%2B39%2F40%29%5E2-1521%2F80%2B8 Now add 8 to both sides to isolate y



y=20%28x%2B39%2F40%29%5E2-881%2F80 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=20, h=-39%2F40, and k=-881%2F80. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=20x%5E2%2B39x%2B8 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C20x%5E2%2B39x%2B8%29 Graph of y=20x%5E2%2B39x%2B8. Notice how the vertex is (-39%2F40,-881%2F80).



Notice if we graph the final equation y=20%28x%2B39%2F40%29%5E2-881%2F80 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C20%28x%2B39%2F40%29%5E2-881%2F80%29 Graph of y=20%28x%2B39%2F40%29%5E2-881%2F80. Notice how the vertex is also (-39%2F40,-881%2F80).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.

Answer by lynnlo(4176) About Me  (Show Source):
You can put this solution on YOUR website!
did you mean to put something in front of your + sign
i did it another way and got
59a+8

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
20a squared + 39a + 8
factor out completely
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f%28a%29+=+20a%5E2+%2B+39a+%2B+8
That can't be factored with integers --> prime
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Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 20x%5E2%2B39x%2B8+=+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=%2839%29%5E2-4%2A20%2A8=881.

Discriminant d=881 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-39%2B-sqrt%28+881+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%2839%29%2Bsqrt%28+881+%29%29%2F2%5C20+=+-0.232958896017209
x%5B2%5D+=+%28-%2839%29-sqrt%28+881+%29%29%2F2%5C20+=+-1.71704110398279

Quadratic expression 20x%5E2%2B39x%2B8 can be factored:
20x%5E2%2B39x%2B8+=+%28x--0.232958896017209%29%2A%28x--1.71704110398279%29
Again, the answer is: -0.232958896017209, -1.71704110398279. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+20%2Ax%5E2%2B39%2Ax%2B8+%29