Chapter 8, Problem 33CT

(a)

To determine

To identify $a$ and $b$

Answer to Problem 33CT

The value of $a$ is $a=\left(x-2\right)$

The value of $b$ is $b=3$

Explanation of Solution

Given Information:

The given expression is ${\left(x-2\right)}^2-9$

The given expression is ${\left(x-2\right)}^2-9$

Rewrite the given expression using $3^2=9$

  ${\left(x-2\right)}^2-3^2$

The given expression is of the from $a^2-b^2$

Comparing $a^2-b^2$ with ${\left(x-2\right)}^2-3^2$ to get $a=\left(x-2\right)$ and $b=3$

(b)

To determine

Factor ${\left(x-2\right)}^2-9$ and simplify.

Answer to Problem 33CT

The factors of ${\left(x-2\right)}^2-3^2=\left(x+2\right)\left(x-5\right)$

Explanation of Solution

Given Information:

The given expression is ${\left(x-2\right)}^2-9$

The given expression is ${\left(x-2\right)}^2-9$

Rewrite the given expression using $3^2=9$

  ${\left(x-2\right)}^2-3^2$

The given expression is of the from $a^2-b^2$

Comparing $a^2-b^2$ with ${\left(x-2\right)}^2-3^2$ to get $a=\left(x-2\right)$ and $b=3$

Formula used: $a^2-b^2=\left(a+b\right)\left(a-b\right)$

Calculations:

  $\begin{array}{c}{\left(x-2\right)}^2-3^2=\left(x-2+3\right)\left(x-2-3\right)\\ =\left(x+2\right)\left(x-5\right)\end{array}$

The factors of ${\left(x-2\right)}^2-3^2=\left(x+2\right)\left(x-5\right)$