SOLUTION: Solve by using the quadratic formula: Z squared + 4z + 1 = 0

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Question 114676: Solve by using the quadratic formula:
Z squared + 4z + 1 = 0
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 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%2B4%2Az%2B1=0 ( notice a=1, b=4, and c=1)




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



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



z+=+%28-4+%2B-+sqrt%28+16%2B-4+%29%29%2F%282%2A1%29 Multiply -4%2A1%2A1 to get -4



z+=+%28-4+%2B-+sqrt%28+12+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



z+=+%28-4+%2B-+2%2Asqrt%283%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



z+=+%28-4+%2B-+2%2Asqrt%283%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

z+=+%28-4+%2B+2%2Asqrt%283%29%29%2F2 or z+=+%28-4+-+2%2Asqrt%283%29%29%2F2


Now break up the fraction


z=-4%2F2%2B2%2Asqrt%283%29%2F2 or z=-4%2F2-2%2Asqrt%283%29%2F2


Simplify


z=-2%2Bsqrt%283%29 or z=-2-sqrt%283%29


So these expressions approximate to

z=-0.267949192431123 or z=-3.73205080756888


So our solutions are:
z=-0.267949192431123 or z=-3.73205080756888

Notice when we graph x%5E2%2B4%2Ax%2B1 (just replace z with x), we get:



when we use the root finder feature on a calculator, we find that x=-0.267949192431123 and x=-3.73205080756888.So this verifies our answer