SOLUTION: Solve the quadratic equation using the Quadratic Formula: 3x^2-2x-2=0

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Question 113993: Solve the quadratic equation using the Quadratic Formula:
3x^2-2x-2=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
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 3%2Ax%5E2-2%2Ax-2=0 ( notice a=3, b=-2, and c=-2)




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



x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%2A3%2A-2+%29%29%2F%282%2A3%29 Negate -2 to get 2



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



x+=+%282+%2B-+sqrt%28+4%2B24+%29%29%2F%282%2A3%29 Multiply -4%2A-2%2A3 to get 24



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



x+=+%282+%2B-+2%2Asqrt%287%29%29%2F%282%2A3%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%282+%2B-+2%2Asqrt%287%29%29%2F6 Multiply 2 and 3 to get 6

So now the expression breaks down into two parts

x+=+%282+%2B+2%2Asqrt%287%29%29%2F6 or x+=+%282+-+2%2Asqrt%287%29%29%2F6


Now break up the fraction


x=%2B2%2F6%2B2%2Asqrt%287%29%2F6 or x=%2B2%2F6-2%2Asqrt%287%29%2F6


Simplify


x=1+%2F+3%2Bsqrt%287%29%2F3 or x=1+%2F+3-sqrt%287%29%2F3


So these expressions approximate to

x=1.21525043702153 or x=-0.548583770354864


So our solutions are:
x=1.21525043702153 or x=-0.548583770354864

Notice when we graph 3%2Ax%5E2-2%2Ax-2, we get:



when we use the root finder feature on a calculator, we find that x=1.21525043702153 and x=-0.548583770354864.So this verifies our answer