Solving Simple Equations Involving Perfect Squares

Here, you will solve simple equations involving perfect squares.
There are two basic approaches you can use to solve an equation like $\,x^2 - 9 = 0\,$:

Approach #1 (factor and use the Zero Factor Law)

To use this approach, you must:

Check that you have $\,0\,$ on one side of the equation.
Factor to get a product on the other side of the equation.
Use the Zero Factor Law:
For all real numbers $\,a\,$ and $\,b\,$, $\,ab = 0\,$ is equivalent to $\,(a = 0\ \ \text{or}\ \ b = 0)\,$.

To use this approach, you would write the following list of equivalent sentences:

$x^2 - 9 = 0$	(original equation; check that $\,0\,$ is on one side of the equation)
$(x+3)(x-3) = 0$	(factor to get a product on the other side)
$x+3 = 0\ \ \text{or}\ \ x-3 = 0$	(use the zero factor law)
$x=-3\ \ \text{or}\ \ x = 3$	(solve the simpler equations)

Approach #2 (use the following theorem)

THEOREM solving equations involving perfect squares

For all real numbers $\,z\ $ and for $\,k\ge 0\,$:

$$ z^2 = k\ \ \ \text{is equivalent to}\ \ \ z=\pm\sqrt{k} $$

Notice that if $\,k\lt 0\,$, then the equation $\,z^2 = k\,$ has no real number solutions.
For example, consider the equation $\,z^2 = -4\,$.
There is no real number which, when squared, gives $\,-4\,$.

To use this approach, you must:

Isolate a perfect square on one side of the equation.
Check that you have a nonnegative number on the other side.
Use the theorem.

To use this approach, you would write the following list of equivalent sentences:

$x^2 - 9 = 0$	(original equation)
$x^2 = 9$	(isolate a perfect square by adding $\,9\,$ to both sides)
$x = \pm\sqrt{9}$	(check that $\,k\ge 0\,$; use the theorem)
$x = \pm 3$	(rename: $\,\sqrt{9} = 3\,$)
$x = 3\ \ \text{or}\ \ x = -3$	(expand the ‘plus or minus’ shorthand, if desired)

EXAMPLES:

Here are three slightly different approaches to solving the equation $\,16x^2 - 25 = 0\,$:

First Approach (use the Zero Factor Law)

$16x^2 - 25 = 0$	(original equation)
$(4x)^2 - 5^2 = 0$	(rewrite left-hand side as a difference of squares)
$(4x + 5)(4x - 5) = 0$	(factor the left-hand side)
$4x + 5 = 0\ \ \text{or}\ \ 4x-5 = 0$	(use the Zero Factor Law)
$4x = -5\ \ \text{or}\ \ 4x = 5$	(solve the simpler equations)
$\displaystyle x = -\frac{5}{4}\ \ \text{or}\ \ x = \frac{5}{4}$	(solve the simpler equations)

Second Approach (use the theorem; isolate $\,x^2\,$)

$16x^2 - 25 = 0$	(original equation)
$16x^2 = 25$	(add $\,25\,$ to both sides)
$\displaystyle x^2 = \frac{25}{16}$	(divide both sides by $\,16\,$: now, $x^2$ is isolated)
$\displaystyle x = \pm\sqrt{\frac{25}{16}}$	(use the theorem)
$\displaystyle x = \pm\frac{5}{4}$	(rename: $\,\sqrt{\frac{25}{16}} = \frac{\sqrt{25}}{\sqrt{16}} = \frac{5}{4}$)
$\displaystyle x = \frac{5}{4}\ \ \text{or}\ \ x = -\frac{5}{4}$	(expand the ‘plus or minus’ shorthand, if desired)

Third Approach (use the theorem; isolate $\,(4x)^2\ $)

$16x^2 - 25 = 0$	(original equation)
$16x^2 = 25$	(add $\,25\,$ to both sides)
$(4x)^2 = 25$	(rename left-hand side as a perfect square)
$4x = \pm\sqrt{25}$	(use the theorem)
$4x = \pm 5$	(rename: $\,\sqrt{25} = 5\,$)
$\displaystyle x = \frac{\pm 5}{4}$	(divide both sides by $\,4\,$)
$\displaystyle x = \frac{5}{4}\ \ \text{or}\ \ x = -\frac{5}{4}$	(expand the ‘plus or minus’ shorthand, if desired)

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Solving More Complicated Equations
Involving Perfect Squares

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.

For more advanced students, a graph is displayed.
For example, the equation $\,x^2 - 9 = 0\,$
is optionally accompanied by the graph of $\,y = x^2 - 9\,$ (the left side of the equation, dashed green)
and the graph of $\,y = 0\,$ (the right side of the equation, solid purple).
In this example, you are finding the values of $\,x\,$ where the green graph intersects the purple graph.
Click the “show/hide graph” button if you prefer not to see the graph.