SOLVING MORE COMPLICATED EQUATIONS INVOLVING PERFECT SQUARES
 

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

As in the previous section, there are two basic approaches you can use.
They're both discussed thoroughly on this page.

The two approaches are illustrated next, by solving the equation $\,(3x+2)^2 = 16\,$.

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

To use this approach, you must:

To use this approach, you would write the following list of equivalent sentences:
[beautiful math coming... please be patient] $\,(3x+2)^2 = 16\,$ (original equation)
[beautiful math coming... please be patient] $\,(3x+2)^2 -16 = 0\,$ (need $\,0\,$ on one side; subtract $\,16\,$ from both sides)
[beautiful math coming... please be patient] $(3x+2)^2 - 4^2 = 0$ (rewrite, so it's clear you have a difference of squares)
[beautiful math coming... please be patient] $(3x+2+4)(3x+2-4) = 0$ (factor the difference of squares)
[beautiful math coming... please be patient] $(3x+6)(3x-2) = 0$ (simplify)
[beautiful math coming... please be patient] $3x+6 = 0\ \ \text{or}\ \ 3x-2 = 0$ (use the zero factor law)
[beautiful math coming... please be patient] $3x = -6\ \ \text{or}\ \ 3x = 2$ (solve the simpler equations)
[beautiful math coming... please be patient] $\displaystyle x = -2\ \ \text{or}\ \ x = \frac{2}{3}$ (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 $\,k\ge 0\,$: [beautiful math coming... please be patient] $$ z^2 = k\ \ \ \text{is equivalent to}\ \ \ z=\pm\sqrt{k} $$

The basic idea is that you're (correctly!) ‘undoing’ a square with the square root.
Notice that if [beautiful math coming... please be patient] $\,k\lt 0\,$, then the equation $\,z^2 = k\,$ has no real number solutions.
For example, consider the equation [beautiful math coming... please be patient] $\,z^2 = -4\,$.
There is no real number which, when squared, gives $\,-4\,$.

To use this approach, you must:

To use this approach, you would write the following list of equivalent sentences:
[beautiful math coming... please be patient] $(3x+2)^2 = 16$ (original equation)
[beautiful math coming... please be patient] $3x + 2 = \pm\sqrt{16}$ (check that $\,k\ge 0\,$; use the theorem)
[beautiful math coming... please be patient] $3x + 2 = \pm 4$ (simplify: $\,\sqrt{16} = 4\,$)
[beautiful math coming... please be patient] $3x + 2 = 4\ \ \text{or}\ \ 3x + 2 = -4$ (expand the ‘plus or minus’ shorthand)
[beautiful math coming... please be patient] $3x = 2\ \ \text{or}\ \ 3x = -6$ (subtract $\,2\,$ from both sides of both equations)
[beautiful math coming... please be patient] $\displaystyle x = \frac{2}{3}\ \ \text{or}\ \ x = -2$ (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

 
 

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.

CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
AVAILABLE MASTERED IN PROGRESS

Solve: