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 $\,(3x+2)^2 = 16\,$.
To use this approach, you must:
[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] $(3x+2+4)(3x+24) = 0$  (factor the difference of squares) 
[beautiful math coming... please be patient] $(3x+6)(3x2) = 0$  (simplify) 
[beautiful math coming... please be patient] $3x+6 = 0\ \ \text{or}\ \ 3x2 = 0$  (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) 
$(3(2)+2)^2\ \ \overset{\text{?}}{=}\ \ 16$  $(3(\frac{2}{3})+2)^2\ \ \overset{\text{?}}{=}\ \ 16$ 
$(6 + 2)^2 \ \ \overset{\text{?}}{=} \ \ 16$  $(2 + 2)^2 \ \ \overset{\text{?}}{=} \ \ 16$ 
$(4)^2 \ \ \overset{\text{?}}{=} \ \ 16$  $(4)^2 \ \ \overset{\text{?}}{=} \ \ 16$ 
$16 = 16$ Check!  $16 = 16$ Check! 
The basic idea is that you're (correctly!) ‘undoing’ a square with the square root.
Notice that if
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$\,k\lt 0\,$,
then the equation $\,z^2 = k\,$ has no real number solutions.
For example, consider the equation
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$\,z^2 = 4\,$.
There is no real number which, when squared, gives $\,4\,$.
To use this approach, you must:
[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 $\,3\,$) 
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:
