In this section, we continue to solve algebraic equations containing the square of our variable x, this time with some fractions involved, just to give you more practice with those and with reducing ratios of square roots. Here's an example:

$$3 x^2 - \frac{2}{3} = \frac{1}{5},$$

As usual for these, remember that

$$\sqrt{x^2} = (\sqrt{x})^2 = x$$

Notice that the squaring and square-root operations (functions, really) undo one another. Applying them in sequence is like doing nothing at all to the variable x. They allow us to "liberate" a variable from a root or square.

The first step to solving such a problem is to do the easy stuff first, namely, move the $-4$ to the right by adding it to both sides:

$$ \begin{align} &3 x^2 - \frac{2}{3} = \frac{1}{5} \\ &\underline{\phantom{000} + \frac{2}{3} + \frac{2}{3}} \\ &\phantom{0000}x^2 = \frac{13}{15} \\[5pt] &\phantom{00000}x = ± \sqrt{\frac{13}{15}} \end{align}$$

Here we can't reduce the roots, but you should always make sure to reduce them to the smallest possible representation of the original(s).

What if   $c - b < 0?$

If $c - b < 0,$ then we end up taking the square root of a negative number. That's possible with a little trick you may or may not know. Check out sections on quadratic functions and complex numbers to learn more, but here's a very short version of how to do it.