SOLUTION: Solve the following quadratic equation. 4x squared = 13x + 12

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Question 114388: Solve the following quadratic equation.
4x squared = 13x + 12
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

4x%5E2=13x%2B12 Start with the given equation


0=13x%2B12-4x%5E2 Move all of the terms to the right side


0=-4x%5E2%2B13x%2B12 Sort the terms


-4x%5E2%2B13x%2B12=0 Switch both sides (I find it easier to read if all of the terms are on the left side)


Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve -4%2Ax%5E2%2B13%2Ax%2B12=0 ( notice a=-4, b=13, and c=12)




x+=+%28-13+%2B-+sqrt%28+%2813%29%5E2-4%2A-4%2A12+%29%29%2F%282%2A-4%29 Plug in a=-4, b=13, and c=12



x+=+%28-13+%2B-+sqrt%28+169-4%2A-4%2A12+%29%29%2F%282%2A-4%29 Square 13 to get 169



x+=+%28-13+%2B-+sqrt%28+169%2B192+%29%29%2F%282%2A-4%29 Multiply -4%2A12%2A-4 to get 192



x+=+%28-13+%2B-+sqrt%28+361+%29%29%2F%282%2A-4%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-13+%2B-+19%29%2F%282%2A-4%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%28-13+%2B-+19%29%2F-8 Multiply 2 and -4 to get -8

So now the expression breaks down into two parts

x+=+%28-13+%2B+19%29%2F-8 or x+=+%28-13+-+19%29%2F-8

Lets look at the first part:

x=%28-13+%2B+19%29%2F-8

x=6%2F-8 Add the terms in the numerator
x=-3%2F4 Divide

So one answer is
x=-3%2F4



Now lets look at the second part:

x=%28-13+-+19%29%2F-8

x=-32%2F-8 Subtract the terms in the numerator
x=4 Divide

So another answer is
x=4

So our solutions are:
x=-3%2F4 or x=4

Notice when we graph -4%2Ax%5E2%2B13%2Ax%2B12, we get:

+graph%28+500%2C+500%2C+-13%2C+14%2C+-13%2C+14%2C-4%2Ax%5E2%2B13%2Ax%2B12%29+

and we can see that the roots are x=-3%2F4 and x=4. This verifies our answer