SOLUTION: use the quadratic formula to solve z^2-8z=-5

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Question 88512: use the quadratic formula to solve z^2-8z=-5
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
z%5E2-8z=-5

z%5E2-8z%2B5=0 Add 5 to both sides

Now let's use the quadratic formula to solve for z:


Starting with the general quadratic

az%5E2%2Bbz%2Bc=0

the general solution using the quadratic equation is:

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

So lets solve z%5E2-8%2Az%2B5=0 ( notice a=1, b=-8, and c=5)

z+=+%28--8+%2B-+sqrt%28+%28-8%29%5E2-4%2A1%2A5+%29%29%2F%282%2A1%29 Plug in a=1, b=-8, and c=5



z+=+%288+%2B-+sqrt%28+%28-8%29%5E2-4%2A1%2A5+%29%29%2F%282%2A1%29 Negate -8 to get 8



z+=+%288+%2B-+sqrt%28+64-4%2A1%2A5+%29%29%2F%282%2A1%29 Square -8 to get 64



z+=+%288+%2B-+sqrt%28+64%2B-20+%29%29%2F%282%2A1%29 Multiply -4%2A5%2A1 to get -20



z+=+%288+%2B-+sqrt%28+44+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



z+=+%288+%2B-+2%2Asqrt%2811%29%29%2F%282%2A1%29 Simplify the square root



z+=+%288+%2B-+2%2Asqrt%2811%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

z+=+%288+%2B+2%2Asqrt%2811%29%29%2F2 or z+=+%288+-+2%2Asqrt%2811%29%29%2F2


Which approximate to

z=7.3166247903554 or z=0.6833752096446


So our solutions are:
z=7.3166247903554 or z=0.6833752096446

Notice when we graph x%5E2-8%2Ax%2B5 (just replace z with x), we get:



when we use the root finder feature on a calculator, we find that x=7.3166247903554 and x=0.6833752096446.So this verifies our answer