SOLUTION: solve simalteneously for x and y: x-y=3 2x^2-3xy+2y(y+1)=3x+13

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Question 726720: solve simalteneously for x and y:
x-y=3
2x^2-3xy+2y(y+1)=3x+13
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
2x^2-3xy+2y(y+1)=3x+13
x-y=3 --> y = x-3
Sub for y in the 1st eqn'
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2x^2-3x(x-3) +2(x-3)*(x-2) = 3x+13
2x%5E2+-+3x%5E2+%2B+9x+%2B+2x%5E2+-+10x+%2B+12+=+3x%2B13
x%5E2+-+x+%2B+12+=+3x%2B13
x%5E2+-+4x+-+1+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-4x%2B-1+=+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=%28-4%29%5E2-4%2A1%2A-1=20.

Discriminant d=20 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--4%2B-sqrt%28+20+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+20+%29%29%2F2%5C1+=+4.23606797749979
x%5B2%5D+=+%28-%28-4%29-sqrt%28+20+%29%29%2F2%5C1+=+-0.23606797749979

Quadratic expression 1x%5E2%2B-4x%2B-1 can be factored:
1x%5E2%2B-4x%2B-1+=+%28x-4.23606797749979%29%2A%28x--0.23606797749979%29
Again, the answer is: 4.23606797749979, -0.23606797749979. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B-1+%29

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x = sqrt(5) + 2, y = sqrt(5) - 1
x =~ 4.236, y = 1.236
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x = 2 - sqrt(5), y = -1 - sqrt(5)
x =~ -0.236, y = -3.236