SOLUTION: Solve the following system of nonlinear equations. 2xy=3 4x^2-8y^2=1

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Question 169249: Solve the following system of nonlinear equations.
2xy=3
4x^2-8y^2=1
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Solve the following system of nonlinear equations.
2xy=3
4x^2-8y^2=1
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From the 1st eqn, x = 3/2y
Sub for x into the 2nd eqn:
4(3/2y)^2 - 8y^2 = 1
9/(y^2) - 8y^2 = 1
Multiply by y^2
9 - 8y^4 = y^2
8y^4 + y^2 - 9 = 0
This is a quadratic in y^2
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 8x%5E2%2B1x%2B-9+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%281%29%5E2-4%2A8%2A-9=289.

Discriminant d=289 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+289+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+289+%29%29%2F2%5C8+=+1
x%5B2%5D+=+%28-%281%29-sqrt%28+289+%29%29%2F2%5C8+=+-1.125

Quadratic expression 8x%5E2%2B1x%2B-9 can be factored:
8x%5E2%2B1x%2B-9+=+%28x-1%29%2A%28x--1.125%29
Again, the answer is: 1, -1.125. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+8%2Ax%5E2%2B1%2Ax%2B-9+%29

Factor:
(y^2 - 1)*(8y^2 + 9) = 0
y^2 = 1, -9/8
y = +1, -1
y = +i*(3/4)sqrt(2)
y = -i*(3/4)sqrt(2) (i = sqrt(-1) )