In this section, we begin to solve algebraic equations containing the square of our variable x,

$$x^2 + a = b,$$

where a and b are constants. Solving these involves taking a square root, the inverse operation of squaring. 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.

Here's an example of this class of problem:

$$x^2 - 4 = 12$$

As usual, 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:

$$ \begin{align} &x^2 - 4 = 12 \\ &\underline{\phantom{00} + 4 \phantom{0} + 4} \\ &\phantom{0000}x^2 = 16 \end{align}$$

Now to find x, we take the square-root of both sides:

$$ \begin{align} \sqrt{x^2} &= ±\sqrt{16}\\ x &= ±4 \end{align}$$

Notice that we get two solutions each time we take a square root because $(4)^2 = 16 \; \color{#E90F89}{\text{ or }} \; (-4)^2 = 16.$ It's important to remember that.

If $b - c < 0,$ 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.

Essentially, we make up a new number, $i = \sqrt{-1}.$ Now the square root of any negative number like -x can be expressed like

$$\sqrt{-x} = \sqrt{-1 \cdot x} = \sqrt{-1}\sqrt{x} = i\sqrt{x}$$

To take the square root of a negative number, simply find the root of the absolute value and append the imaginary number, $i.$

You can practice problems of this type below. They are generated randomly. Practice them until you consistently get them right; these should eventually be very easy for you.

Hit answer to view solution.

In this section, we'll refer often to **inverse** operations. Inverse operations are opposite, and one can be used to *undo* the action of the other.

**Addition**and**subtraction**are inverse operations.**Multiplication**and**division**are inverse operations.- The square-root and squaring functions or operations are inverses. Each undoes the action of the other.

There are a number of these pages you can use for algebra practice. Just pick the rough type of problem you need to work on.

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