Question

Is it possible for two different numbers when squared to give the same result? What does this result tell you about solving an equation when the variable is squared? How many solutions will an equations like this have? Will there always be the same number of solutions for any equation with a squared wvariable? Explain

Answer 1

Yes, is it possible. Since the product of two negative numbers is a positive numbers, for any real number x you have

$x \cdot x = x^2,\text{ but also }\ (-x)(-x) = x^2$

So, a number and its opposite give the same result when squared. For example, both 3 and -3 give 9 when squared.

So, when a variable is squared into an equation, you will have two different solutions. For example, in the equation

$x^2 = 25$

You are looking for a number that gives 25 when squared. We know that 5 gives 25 when squared, but because of everything we said at the beginning, -5 also gives 25 when squared. So, the two solutions are 5 and -5.

About the last question, I assume that by "an equation with a squared variable" you mean a quadratic equation, i.e. something like

$ax^2+bx+c=0$

In this case, there can be two different solutions, one double solution, or no solution at all. In fact, the solving formula involves a square root, namely

$\sqrt{b^2-4ac}$

and a root doesn't always exist.