Solving problems of the type   $x^2 + a = b$


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.

What if   $b - a < 0?$

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.

The imaginary number   $i$

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.

Inverse operations

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.

Other Algebra practice problems

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.

Type 1:   $ax + by = c$

Type 2:   $\frac{a}{b}x + \frac{c}{d} = \frac{e}{f}$

Type 3:   $\frac{a}{x} + b = c$

Type 4:   $\frac{a}{x} + \frac{b}{c} = \frac{d}{e}$

Type 5:   $x^2 + b = c$

Type 6:   $ax^2 + b = c$

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