Follow up to Angie's question?

Question

If both sides are not = then why is there an equal sign? And is the answer 2 or both 2 and -1? Thanks!

Answer 1

The quadratic equation yields two solutions:

x = 2 and x = -1

But that quadratic equation is not equivalent to the original equation. What you know for sure is that any solution of the original equation will also satisfy the quadratic equation. That means that the quadratic equation can have more solutions that the original equation has.

You can check that x = 2 is a solution and that x = -1 is not a solution.

A deeper reason why you got the extra solution is that the square root function is defined to be the positive inverse of the function y = x^2. This function has two inverses that are negatives of each other.

When you obtained the quadratic equation by squaring both sides, you only used the fact that the square root is an inverse of the quadratic function, not that it is the positve inverse. This means that had we defined the square root functon as the negative inverse, we would have obtained the same quadratic equation. The solution of the quadratic equation will thus also contain the solution of the equation with the square root replaced by its negative.

Let's check this:

-sqrt(3-x) for x = -1 is equal to -2

x - 1 for x = -1 is equal to -2.

Answer 2

Thank you Count Iblis for the great explanation and confirmation of my objection to that post.

This is in reference to
http://www.jiskha.com/display.cgi?id=1247010204