Algebra Examples

$\frac{p^{2} + 2 p - 51}{p^{2} - 6 p - 27} - \frac{4}{p + 3}$

Step 1

Factor $p^{2} - 6 p - 27$ 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 $- 27$ and whose sum is $- 6$ .

$- 9, 3$

Step 1.2

Write the factored form using these integers.

$\frac{p^{2} + 2 p - 51}{(p - 9) (p + 3)} - \frac{4}{p + 3}$

Step 2

To write $- \frac{4}{p + 3}$ as a fraction with a common denominator, multiply by $\frac{p - 9}{p - 9}$ .

$\frac{p^{2} + 2 p - 51}{(p - 9) (p + 3)} - \frac{4}{p + 3} \cdot \frac{p - 9}{p - 9}$

Step 3

Write each expression with a common denominator of $(p - 9) (p + 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{4}{p + 3}$ by $\frac{p - 9}{p - 9}$ .

$\frac{p^{2} + 2 p - 51}{(p - 9) (p + 3)} - \frac{4 (p - 9)}{(p + 3) (p - 9)}$

Step 3.2

Reorder the factors of $(p - 9) (p + 3)$ .

$\frac{p^{2} + 2 p - 51}{(p + 3) (p - 9)} - \frac{4 (p - 9)}{(p + 3) (p - 9)}$

Step 4

Combine the numerators over the common denominator.

$\frac{p^{2} + 2 p - 51 - 4 (p - 9)}{(p + 3) (p - 9)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{p^{2} + 2 p - 51 - 4 p - 4 \cdot - 9}{(p + 3) (p - 9)}$

Step 5.2

Multiply $- 4$ by $- 9$ .

$\frac{p^{2} + 2 p - 51 - 4 p + 36}{(p + 3) (p - 9)}$

Step 5.3

Subtract $4 p$ from $2 p$ .

$\frac{p^{2} - 2 p - 51 + 36}{(p + 3) (p - 9)}$

Step 5.4

Add $- 51$ and $36$ .

$\frac{p^{2} - 2 p - 15}{(p + 3) (p - 9)}$

Step 5.5

Factor $p^{2} - 2 p - 15$ using the AC method.

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Step 5.5.1

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 $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 5.5.2

Write the factored form using these integers.

$\frac{(p - 5) (p + 3)}{(p + 3) (p - 9)}$

Step 6

Cancel the common factor of $p + 3$ .

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Step 6.1

Cancel the common factor.

$\frac{(p - 5) (p + 3)}{(p + 3) (p - 9)}$

Step 6.2

Rewrite the expression.

$\frac{p - 5}{p - 9}$