SOLUTION: x^2-x-13=0

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Question 115710This question is from textbook
: x^2-x-13=0 This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
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-x-13=0 ( notice a=1, b=-1, and c=-13)




x+=+%28--1+%2B-+sqrt%28+%28-1%29%5E2-4%2A1%2A-13+%29%29%2F%282%2A1%29 Plug in a=1, b=-1, and c=-13



x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%2A1%2A-13+%29%29%2F%282%2A1%29 Negate -1 to get 1



x+=+%281+%2B-+sqrt%28+1-4%2A1%2A-13+%29%29%2F%282%2A1%29 Square -1 to get 1 (note: remember when you square -1, you must square the negative as well. This is because %28-1%29%5E2=-1%2A-1=1.)



x+=+%281+%2B-+sqrt%28+1%2B52+%29%29%2F%282%2A1%29 Multiply -4%2A-13%2A1 to get 52



x+=+%281+%2B-+sqrt%28+53+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%281+%2B-+sqrt%2853%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%281+%2B-+sqrt%2853%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%281+%2B+sqrt%2853%29%29%2F2 or x+=+%281+-+sqrt%2853%29%29%2F2


Now break up the fraction


x=%2B1%2F2%2Bsqrt%2853%29%2F2 or x=%2B1%2F2-sqrt%2853%29%2F2


Simplify


x=1+%2F+2%2Bsqrt%2853%29%2F2 or x=1+%2F+2-sqrt%2853%29%2F2


So these expressions approximate to

x=4.14005494464026 or x=-3.14005494464026


So our solutions are:
x=4.14005494464026 or x=-3.14005494464026

Notice when we graph x%5E2-x-13, we get:



when we use the root finder feature on a calculator, we find that x=4.14005494464026 and x=-3.14005494464026.So this verifies our answer