SOLUTION: Solve the quadratic equation using the Quadratic Formula: 2y^2-y=6

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Question 113992: Solve the quadratic equation using the Quadratic Formula:
2y^2-y=6
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
2y%5E2-y=6 Start with the given equation


2y%5E2-y-6=0 Subtract 6 from both sides

Let's use the quadratic formula to solve for y:


Starting with the general quadratic

ay%5E2%2Bby%2Bc=0

the general solution using the quadratic equation is:

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



So lets solve 2%2Ay%5E2-y-6=0 ( notice a=2, b=-1, and c=-6)




y+=+%28--1+%2B-+sqrt%28+%28-1%29%5E2-4%2A2%2A-6+%29%29%2F%282%2A2%29 Plug in a=2, b=-1, and c=-6



y+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%2A2%2A-6+%29%29%2F%282%2A2%29 Negate -1 to get 1



y+=+%281+%2B-+sqrt%28+1-4%2A2%2A-6+%29%29%2F%282%2A2%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.)



y+=+%281+%2B-+sqrt%28+1%2B48+%29%29%2F%282%2A2%29 Multiply -4%2A-6%2A2 to get 48



y+=+%281+%2B-+sqrt%28+49+%29%29%2F%282%2A2%29 Combine like terms in the radicand (everything under the square root)



y+=+%281+%2B-+7%29%2F%282%2A2%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



y+=+%281+%2B-+7%29%2F4 Multiply 2 and 2 to get 4

So now the expression breaks down into two parts

y+=+%281+%2B+7%29%2F4 or y+=+%281+-+7%29%2F4

Lets look at the first part:

x=%281+%2B+7%29%2F4

y=8%2F4 Add the terms in the numerator
y=2 Divide

So one answer is
y=2



Now lets look at the second part:

x=%281+-+7%29%2F4

y=-6%2F4 Subtract the terms in the numerator
y=-3%2F2 Divide

So another answer is
y=-3%2F2

So our solutions are:
y=2 or y=-3%2F2

Notice when we graph 2%2Ax%5E2-x-6 (just replace y with x), we get:

+graph%28+500%2C+500%2C+-13%2C+12%2C+-13%2C+12%2C2%2Ax%5E2%2B-1%2Ax%2B-6%29+

and we can see that the roots are y=2 and y=-3%2F2. This verifies our answer