Pre-Algebra Examples

$\frac{1}{r^{2} - 1} = \frac{2}{r^{2} + r - 2}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\frac{1}{r^{2} - 1^{2}} = \frac{2}{r^{2} + r - 2}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r$ and $b = 1$ .

$\frac{1}{(r + 1) (r - 1)} = \frac{2}{r^{2} + r - 2}$

Step 2

Factor $r^{2} + r - 2$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 2.2

Write the factored form using these integers.

$\frac{1}{(r + 1) (r - 1)} = \frac{2}{(r - 1) (r + 2)}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$r = 0, 1$

Step 4