SOLUTION: y^2-12y+36=49 c^2+16c+64=15

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Question 112305: y^2-12y+36=49
c^2+16c+64=15
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
y%5E2-12y%2B36=49 Start with the given equation


y%5E2-12y-13=0 Subtract 49 from both sides

%28y-13%29%28y%2B1%29=0 Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:

y-13=0 or y%2B1=0

y=13 or y=-1 Now solve for y in each case


So our solutions are y=13 or y=-1


Notice if we graph y=x%5E2-12x-13 (just replace y with x) we get

+graph%28500%2C500%2C-10%2C15%2C-10%2C10%2C+x%5E2-12x-13%29+

and we can see that the graph has roots at y=13 and y=-1, so this verifies our answer.




#2


c%5E2%2B16c%2B64=15 Start with the given equation


c%5E2%2B16c%2B49=0 Subtract 15 from both sides


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


Starting with the general quadratic

ac%5E2%2Bbc%2Bc=0

the general solution using the quadratic equation is:

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



So lets solve c%5E2%2B16%2Ac%2B49=0 ( notice a=1, b=16, and c=49)




c+=+%28-16+%2B-+sqrt%28+%2816%29%5E2-4%2A1%2A49+%29%29%2F%282%2A1%29 Plug in a=1, b=16, and c=49



c+=+%28-16+%2B-+sqrt%28+256-4%2A1%2A49+%29%29%2F%282%2A1%29 Square 16 to get 256



c+=+%28-16+%2B-+sqrt%28+256%2B-196+%29%29%2F%282%2A1%29 Multiply -4%2A49%2A1 to get -196



c+=+%28-16+%2B-+sqrt%28+60+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



c+=+%28-16+%2B-+2%2Asqrt%2815%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



c+=+%28-16+%2B-+2%2Asqrt%2815%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

c+=+%28-16+%2B+2%2Asqrt%2815%29%29%2F2 or c+=+%28-16+-+2%2Asqrt%2815%29%29%2F2


Now break up the fraction


c=-16%2F2%2B2%2Asqrt%2815%29%2F2 or c=-16%2F2-2%2Asqrt%2815%29%2F2


Simplify


c=-8%2Bsqrt%2815%29 or c=-8-sqrt%2815%29


So these expressions approximate to

c=-4.12701665379258 or c=-11.8729833462074


So our solutions are:
c=-4.12701665379258 or c=-11.8729833462074

Notice when we graph x%5E2%2B16%2Ax%2B49 (just replace c with x), we get:



when we use the root finder feature on a calculator, we find that x=-4.12701665379258 and x=-11.8729833462074.So this verifies our answer