SOLUTION: solve the equation by using the quadratic formula 3z^2-8z+2=0

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Question 94385: solve the equation by using the quadratic formula
3z^2-8z+2=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 3%2Az%5E2-8%2Az%2B2=0 ( notice a=3, b=-8, and c=2)

z+=+%28--8+%2B-+sqrt%28+%28-8%29%5E2-4%2A3%2A2+%29%29%2F%282%2A3%29 Plug in a=3, b=-8, and c=2



z+=+%288+%2B-+sqrt%28+%28-8%29%5E2-4%2A3%2A2+%29%29%2F%282%2A3%29 Negate -8 to get 8



z+=+%288+%2B-+sqrt%28+64-4%2A3%2A2+%29%29%2F%282%2A3%29 Square -8 to get 64 (note: remember when you square -8, you must square the negative as well. This is because %28-8%29%5E2=-8%2A-8=64.)



z+=+%288+%2B-+sqrt%28+64%2B-24+%29%29%2F%282%2A3%29 Multiply -4%2A2%2A3 to get -24



z+=+%288+%2B-+sqrt%28+40+%29%29%2F%282%2A3%29 Combine like terms in the radicand (everything under the square root)



z+=+%288+%2B-+2%2Asqrt%2810%29%29%2F%282%2A3%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



z+=+%288+%2B-+2%2Asqrt%2810%29%29%2F6 Multiply 2 and 3 to get 6

So now the expression breaks down into two parts

z+=+%288+%2B+2%2Asqrt%2810%29%29%2F6 or z+=+%288+-+2%2Asqrt%2810%29%29%2F6


Now break up the fraction


z=%2B8%2F6%2B2%2Asqrt%2810%29%2F6 or z=%2B8%2F6-2%2Asqrt%2810%29%2F6


Simplify


z=4+%2F+3%2Bsqrt%2810%29%2F3 or z=4+%2F+3-sqrt%2810%29%2F3


So these expressions approximate to

z=2.38742588672279 or z=0.279240779943873


So our solutions are:
z=2.38742588672279 or z=0.279240779943873

Notice when we graph 3%2Ax%5E2-8%2Ax%2B2 (just replace z with x), we get:



when we use the root finder feature on a calculator, we find that x=2.38742588672279 and x=0.279240779943873.So this verifies our answer