Pre-Algebra Examples

$\frac{4}{p^{2} + 7 p - 18} = \frac{6}{p + 9} - \frac{4}{p - 2}$

Step 1

Factor $p^{2} + 7 p - 18$ using the AC method.

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Step 1.1

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 $- 18$ and whose sum is $7$ .

$- 2, 9$

Step 1.2

Write the factored form using these integers.

$\frac{4}{(p - 2) (p + 9)} = \frac{6}{p + 9} - \frac{4}{p - 2}$

Step 2

Simplify $\frac{6}{p + 9} - \frac{4}{p - 2}$ .

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Step 2.1

To write $\frac{6}{p + 9}$ as a fraction with a common denominator, multiply by $\frac{p - 2}{p - 2}$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{6}{p + 9} \cdot \frac{p - 2}{p - 2} - \frac{4}{p - 2}$

Step 2.2

To write $- \frac{4}{p - 2}$ as a fraction with a common denominator, multiply by $\frac{p + 9}{p + 9}$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{6}{p + 9} \cdot \frac{p - 2}{p - 2} - \frac{4}{p - 2} \cdot \frac{p + 9}{p + 9}$

Step 2.3

Write each expression with a common denominator of $(p + 9) (p - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{6}{p + 9}$ by $\frac{p - 2}{p - 2}$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{6 (p - 2)}{(p + 9) (p - 2)} - \frac{4}{p - 2} \cdot \frac{p + 9}{p + 9}$

Step 2.3.2

Multiply $\frac{4}{p - 2}$ by $\frac{p + 9}{p + 9}$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{6 (p - 2)}{(p + 9) (p - 2)} - \frac{4 (p + 9)}{(p - 2) (p + 9)}$

Step 2.3.3

Reorder the factors of $(p - 2) (p + 9)$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{6 (p - 2)}{(p + 9) (p - 2)} - \frac{4 (p + 9)}{(p + 9) (p - 2)}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{4}{(p - 2) (p + 9)} = \frac{6 (p - 2) - 4 (p + 9)}{(p + 9) (p - 2)}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Factor $2$ out of $6 (p - 2) - 4 (p + 9)$ .

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Step 2.5.1.1

Factor $2$ out of $6 (p - 2)$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{2 (3 (p - 2)) - 4 (p + 9)}{(p + 9) (p - 2)}$

Step 2.5.1.2

Factor $2$ out of $- 4 (p + 9)$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{2 (3 (p - 2)) + 2 (- 2 (p + 9))}{(p + 9) (p - 2)}$

Step 2.5.1.3

Factor $2$ out of $2 (3 (p - 2)) + 2 (- 2 (p + 9))$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{2 (3 (p - 2) - 2 (p + 9))}{(p + 9) (p - 2)}$

Step 2.5.2

Apply the distributive property.

$\frac{4}{(p - 2) (p + 9)} = \frac{2 (3 p + 3 \cdot - 2 - 2 (p + 9))}{(p + 9) (p - 2)}$

Step 2.5.3

Multiply $3$ by $- 2$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{2 (3 p - 6 - 2 (p + 9))}{(p + 9) (p - 2)}$

Step 2.5.4

Apply the distributive property.

$\frac{4}{(p - 2) (p + 9)} = \frac{2 (3 p - 6 - 2 p - 2 \cdot 9)}{(p + 9) (p - 2)}$

Step 2.5.5

Multiply $- 2$ by $9$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{2 (3 p - 6 - 2 p - 18)}{(p + 9) (p - 2)}$

Step 2.5.6

Subtract $2 p$ from $3 p$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{2 (p - 6 - 18)}{(p + 9) (p - 2)}$

Step 2.5.7

Subtract $18$ from $- 6$ .

$\frac{4}{(p - 2) (p + 9)} = \frac{2 (p - 24)}{(p + 9) (p - 2)}$

Step 3

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

$p \approx 26$

Step 4