SOLUTION: Find the exact solutions to {{{x^2+4x+4=16}}}

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Question 130756: Find the exact solutions to x%5E2%2B4x%2B4=16
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2B4x%2B4=16 Start with the given equation


x%5E2%2B4x%2B4-16=0 Subtract 16 from both sides.


x%5E2%2B4x-12=0 Combine like terms



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 x%5E2%2B4%2Ax-12=0 ( notice a=1, b=4, and c=-12)




x+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A1%2A-12+%29%29%2F%282%2A1%29 Plug in a=1, b=4, and c=-12



x+=+%28-4+%2B-+sqrt%28+16-4%2A1%2A-12+%29%29%2F%282%2A1%29 Square 4 to get 16



x+=+%28-4+%2B-+sqrt%28+16%2B48+%29%29%2F%282%2A1%29 Multiply -4%2A-12%2A1 to get 48



x+=+%28-4+%2B-+sqrt%28+64+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%28-4+%2B-+8%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%28-4+%2B-+8%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%28-4+%2B+8%29%2F2 or x+=+%28-4+-+8%29%2F2

Lets look at the first part:

x=%28-4+%2B+8%29%2F2

x=4%2F2 Add the terms in the numerator
x=2 Divide

So one answer is
x=2



Now lets look at the second part:

x=%28-4+-+8%29%2F2

x=-12%2F2 Subtract the terms in the numerator
x=-6 Divide

So another answer is
x=-6




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Answer:

So the exact solutions are:

x=-6 or x=2



Notice when we graph x%5E2%2B4%2Ax-12, we get:

+graph%28+500%2C+500%2C+-16%2C+12%2C+-16%2C+12%2C1%2Ax%5E2%2B4%2Ax%2B-12%29+

and we can see that the roots are x=-6 and x=2. This verifies our answer