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

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: