Solving More Complicated Equations Involving Perfect Squares

Here, you will solve more complicated equations involving perfect squares.

There are two basic approaches you can use to solve an equation like $\,(3x+2)^2 = 16\,$:

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

To use this approach, you must:

Get $\,0\,$ 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:

$\,(3x+2)^2 = 16\,$	(original equation)
$\,(3x+2)^2 -16 = 0\,$	(need $\,0\,$ on one side; subtract $\,16\,$ from both sides)
$(3x+2)^2 - 4^2 = 0$	(rewrite, so it's clear you have a difference of squares)
$(3x+2+4)(3x+2-4) = 0$	(factor the difference of squares)
$(3x+6)(3x-2) = 0$	(simplify)
$3x+6 = 0\ \ \text{or}\ \ 3x-2 = 0$	(use the zero factor law)
$3x = -6\ \ \text{or}\ \ 3x = 2$	(solve the simpler equations)
$\displaystyle x = -2\ \ \text{or}\ \ x = \frac{2}{3}$	(solve the simpler equations)

It's a good idea to check:

$(3(-2)+2)^2\ \ \overset{\text{?}}{=}\ \ 16$	$(3(\frac{2}{3})+2)^2\ \ \overset{\text{?}}{=}\ \ 16$
$(-6 + 2)^2 \ \ \overset{\text{?}}{=} \ \ 16$	$(2 + 2)^2 \ \ \overset{\text{?}}{=} \ \ 16$
$(-4)^2 \ \ \overset{\text{?}}{=} \ \ 16$	$(4)^2 \ \ \overset{\text{?}}{=} \ \ 16$
$16 = 16$ Check!	$16 = 16$ Check!

Approach #2 (use the following theorem)

THEOREM solving equations involving perfect squares

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

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

The basic idea is that you're (correctly!) ‘undoing’ a square with the square root.
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:

$(3x+2)^2 = 16$	(original equation)
$3x + 2 = \pm\sqrt{16}$	(check that $\,k\ge 0\,$; use the theorem)
$3x + 2 = \pm 4$	(simplify: $\,\sqrt{16} = 4\,$)
$3x + 2 = 4\ \ \text{or}\ \ 3x + 2 = -4$	(expand the ‘plus or minus’ shorthand)
$3x = 2\ \ \text{or}\ \ 3x = -6$	(subtract $\,2\,$ from both sides of both equations)
$\displaystyle x = \frac{2}{3}\ \ \text{or}\ \ x = -2$	(divide both sides of both equations by $\,3\,$)

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

When you're done practicing, move on to:
Translating Simple Mathematical Phrases

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 $\,(3x+2)^2 = 16\,$
is optionally accompanied by the graph of $\,y = (3x+2)^2\,$ (the left side of the equation, dashed green)
and the graph of $\,y = 16\,$ (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.