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.
(