Chapter 3.1, Problem 72E

To determine

To factor the given expression.

Answer to Problem 72E

The factors of $x^2-4$ is $\left(x-2\right)\left(x+2\right)$ and factors of $x^2-9$ is $\left(x-3\right)\left(x+3\right)$ .

The $x$ -intercepts of $y=x^2-a^2$ are $x=-a$ and $x=a$ with vertex at $\left(0,-a^2\right)$ and axis of symmetry is $y$ -axis.

Explanation of Solution

Given information:

Factor the expressions $x^2-4$ and $x^2-9$ . Use it to factor $x^2-a^2$ and graph the function $y=x^2-a^2$ and label the vertex, $x$ -intercepts and the axis of symmetry.

Concept used:

We have, $a^2-b^2=\left(a-b\right)\left(a+b\right)$

Calculation:

  $\begin{array}{c}{x}^2-4=x^2-2^2\\ =\left(x-2\right)\left(x+2\right)\end{array}$

Similarly,

  $\begin{array}{c}{x}^2-9=x^2-3^2\\ =\left(x-3\right)\left(x+3\right)\end{array}$

Thus, $x^2-a^2=\left(x-a\right)\left(x+a\right)$ .

Graph the function $y=x^2-a^2$ .

The graph will cut the $x$ -axis when $y=0$ .

  $\begin{array}{c}{x}^2-a^2=0\\ \left(x-a\right)\left(x+a\right)=0\\ x=-a,x=a\end{array}$

Thus, the $x$ -intercepts are $x=-a$ and $x=a$ .

When $x=0$ , $y=-a^2$ , thus the graph cut the $y$ -axis at $\left(0,-a^2\right)$ .

  

Clearly, the graph is an upward parabola with vertex at $\left(0,-a^2\right)$ and axis of symmetry $y$ -axis since the graph is symmetrical about $y$ -axis.