WORD PROBLEMS INVOLVING PERFECT SQUARES

Here, you will solve word problems that result in equations involving perfect squares.

EXAMPLES:
Question: I'm thinking of a number.
The square of $\,3\,$ times this number is $\,25\,$.
What number(s) could I be thinking of?
Solution:

$(3x)^2 = 25$

$3x = \pm 5$

$3x = 5\ \ \text{or}\ \ 3x = -5$

$x = \frac{5}{3}\ \ \text{or}\ \ x = -\frac{5}{3}$
Question: I'm thinking of a number.
When I take one less than three times this number, and then square the result, I end up with the number $\,25\,$.
What number(s) could I be thinking of?
Solution:

$(3x-1)^2 = 25$

$3x-1 = \pm 5$

$3x-1 = 5\ \ \text{or}\ \ 3x-1 = -5$

$3x = 6\ \ \text{or}\ \ 3x = -4$

$x = 2\ \ \text{or}\ \ x = -\frac{4}{3}$
Question: I'm thinking of a negative number.
When I take the sum of this number and $\,2\,$, and then square the result, I end up with the number $\,9\,$.
What number am I thinking of?
Solution:

$(x+2)^2 = 9$

$x+2 = \pm 3$

$x+2 = 3\ \ \text{or}\ \ x+2 = -3$

$x = 1\ \ \text{or}\ \ x = -5$

Since the number being thought of is negative, the answer is $\,-5\,$.
Question: I'm thinking of a positive number.
When I take the difference of this number and $\,3\,$, and then square the result, I end up with the number $\,16\,$.
What number am I thinking of?
Solution:

$(x-3)^2 = 16$

$x-3 = \pm 4$

$x-3 = 4\ \ \text{or}\ \ x-3 = -4$

$x = 7\ \ \text{or}\ \ x = -1$

Since the number being thought of is positive, the answer is $\,7\,$.
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Basic Concepts Involved in Factoring Trinomials

CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
It is possible that there are no numbers that work.
PROBLEM TYPES:
 1 2 3 4 5 6 7 8 9 10 11 12
AVAILABLE MASTERED IN PROGRESS
 Solve this word problem:
 (MAX is 12; there are 12 different problem types.)