SOLUTION: Please help me solve this equation: {{{x^2+x+ sqrt(x^2+x)-2=0}}}

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Question 315243: Please help me solve this equation: x%5E2%2Bx%2B+sqrt%28x%5E2%2Bx%29-2=0
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2%2Bx%2B+sqrt%28x%5E2%2Bx%29-2=0
x%5E2%2Bx-2=-sqrt%28x%5E2%2Bx%29
Square both sides,
x%5E4%2B2x%5E3-3x%5E2-4x%2B4=x%5E2%2Bx
x%5E4%2B2x%5E3-4x%5E2-5x%2B4=0
You can factor this equation into the product of two quadratics,
%28x%5E2%2Bx-4%29%28x%5E2%2Bx-1%29=0
Use the quadratic formula to find the roots of each quadratic.
First solution:x%5E2%2Bx-4
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A1%2A%28-4%29+%29%29%2F%282%2A1%29+
x+=+%28-1+%2B-+sqrt%28+17%29%29%2F2+
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Second solution:x%5E2%2Bx-1
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A1%2A%28-1%29+%29%29%2F%282%2A1%29+
x+=+%28-1+%2B-+sqrt%28+5%29%29%2F2+
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Verify the solutions.
The first set of solutions does not satisfy the equation for reasons having to do with the +/- feature relating to the square root.
sqrt%28a%5E2%29=0+%2B-+a
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The second set of solutions does satisfy the equation.
highlight%28x+=+%28-1+%2B-+sqrt%28+5%29%29%2F2%29+ or approximately,
x=0.611 and x=-1.611
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We can also verify by graphing the function, x%5E2%2Bx%2B+sqrt%28x%5E2%2Bx%29-2 and noting where it crosses the x-axis.
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+graph%28+300%2C+300%2C+-2%2C+2%2C+-2%2C+2%2C+x%5E2%2Bx%2B+sqrt%28x%5E2%2Bx%29-2%29+