document.write( "Question 207916: Equations Quadratic in form\r
\n" ); document.write( "\n" ); document.write( "X^2 + x + (sqrt x^2 + x)- 2 = 0
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Algebra.Com's Answer #157280 by jsmallt9(3758) $\"\"$ $\"About$

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Quadratic equations are of the form:
\n" ); document.write( " $\"ax%5E2+%2B+bx+%2B+c+=+0\"$
\n" ); document.write( "At first glance
\n" ); document.write( " $\"x%5E2+%2B+x+%2B+sqrt%28x%5E2+%2B+x%29-+2+=+0\"$
\n" ); document.write( "does not appear to be of the quadratic form. But a deeper look reveals that it is a quadratic equation. The key is to notice that

The expression $\"x%5E2%2Bx\"$ appears twice; and
$\"x%5E2%2Bx+=+%28sqrt%28x%5E2%2Bx%29%29%5E2\"$

\n" ); document.write( "With this in mind we can use a substitution to see and solve the quadratic equation.

\n" ); document.write( "Let's make $\"y+=+sqrt%28x%5E2%2Bx%29\"$ . Then $\"y%5E2+=+%28sqrt%28x%5E2%2Bx%29%29%5E2+=+x%5E2+%2B+x\"$ . Substituting these into the original equation we get:
\n" ); document.write( " $\"y%5E2+%2B+y+-+2+=+0\"$
\n" ); document.write( "We can solve this for y by factoring (or with the quadratic formula):
\n" ); document.write( " $\"%28y+%2B+2%29%28y+-+1%29+=+0\"$
\n" ); document.write( "The only way this product can be zero is if one of the products is zero:
\n" ); document.write( " $\"y+%2B+2+=+0\"$ or $\"y+-+1+=+0\"$
\n" ); document.write( "Solving each of these we get:
\n" ); document.write( " $\"y+=+-2\"$ or $\"y+=+1\"$
\n" ); document.write( "Substituting back in for the y's we get:
\n" ); document.write( " $\"sqrt%28x%5E2%2Bx%29+=+-2\"$ or $\"sqrt%28x%5E2%2Bx%29+=+1\"$
\n" ); document.write( "Since all square roots are positive it is impossible, in the first equation, for the square root to be a -2. So there are no solutions to the first equation. But there are solutions to the second equation. Square both sides to get:
\n" ); document.write( " $\"%28sqrt%28x%5E2+%2B+x%29%29%5E2+=+1%5E2\"$
\n" ); document.write( " $\"x%5E2+%2B+x+=+1\"$
\n" ); document.write( "Subtracting one from both sides we get:
\n" ); document.write( " $\"x%5E2+%2B+x+-+1+=+0\"$
\n" ); document.write( "This does not factor but we can use the quadratic formula on it:
\n" ); document.write( " $\"x+=+%28-1+%2B-+sqrt%281%5E2+-+4%281%29%28-1%29%29%29%2F%282%281%29%29\"$
\n" ); document.write( "Simplifying we get:
\n" ); document.write( " $\"x+=+%28-1+%2B-+sqrt%281+-+4%281%29%28-1%29%29%29%2F%282%281%29%29\"$
\n" ); document.write( " $\"x+=+%28-1+%2B-+sqrt%281+-+%28-4%29%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28-1+%2B-+sqrt%285%29%29%2F2\"$
\n" ); document.write( "Since we squared both sides while solving, we should check for extraneous solutions. First we'll check $\"x+=+%28-1%2Bsqrt%285%29%29%2F2\"$ . If we square this we get:
\n" ); document.write( " $\"x%5E2+=+%286+-+2sqrt%285%29%29%2F4\"$
\n" ); document.write( "Add x to both sides:
\n" ); document.write( " $\"x%5E2+%2B+x+=+%286+-+2sqrt%285%29%29%2F4+%2B+%28-1%2Bsqrt%285%29%29%2F2\"$
\n" ); document.write( "Match the denominators so we can add:
\n" ); document.write( " $\"x%5E2+%2B+x+=+%286+-+2sqrt%285%29%29%2F4+%2B+%28-2+%2B+2sqrt%285%29%29%2F4\"$
\n" ); document.write( " $\"x%5E2+%2B+x+=+4%2F4\"$
\n" ); document.write( " $\"x%5E2+%2B+x+=+1\"$
\n" ); document.write( "Substituting into the original equation we get:
\n" ); document.write( " $\"1+%2B+sqrt%281%29+-+2+=+0\"$
\n" ); document.write( "which checks out.

\n" ); document.write( "Checking the other solution, $\"x+=+%28-1+-+sqrt%285%29%29%2F2\"$ , in a similar way finds that it, too, works. So both $\"x+=+%28-1+%2B+sqrt%285%29%29%2F2\"$ and $\"x+=+%28-1+-+sqrt%285%29%29%2F2\"$ are solutions. \n" ); document.write( "

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