SOLUTION: Equations Quadratic in form X^2 + x + (sqrt x^2 + x)- 2 = 0

Question 207916: Equations Quadratic in form
X^2 + x + (sqrt x^2 + x)- 2 = 0

Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Quadratic equations are of the form:

At first glance

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 appears twice; and

With this in mind we can use a substitution to see and solve the quadratic equation.

Let's make . Then . Substituting these into the original equation we get:

We can solve this for y by factoring (or with the quadratic formula):

The only way this product can be zero is if one of the products is zero:
or
Solving each of these we get:
or
Substituting back in for the y's we get:
or
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:

Subtracting one from both sides we get:

This does not factor but we can use the quadratic formula on it:

Simplifying we get:

Since we squared both sides while solving, we should check for extraneous solutions. First we'll check . If we square this we get:

Add x to both sides:

Match the denominators so we can add:

Substituting into the original equation we get:

which checks out.

Checking the other solution, , in a similar way finds that it, too, works. So both and are solutions.
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