Algebra Examples

$\frac{x^{2} + 5 x - 36}{x^{2} - 49} \cdot (x^{2} - 11 x + 28)$

Step 1

Factor $x^{2} + 5 x - 36$ 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 $- 36$ and whose sum is $5$ .

$- 4, 9$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 9)}{x^{2} - 49} \cdot (x^{2} - 11 x + 28)$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $49$ as $7^{2}$ .

$\frac{(x - 4) (x + 9)}{x^{2} - 7^{2}} \cdot (x^{2} - 11 x + 28)$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$\frac{(x - 4) (x + 9)}{(x + 7) (x - 7)} \cdot (x^{2} - 11 x + 28)$

Step 3

Multiply $\frac{(x - 4) (x + 9)}{(x + 7) (x - 7)}$ by $x^{2} - 11 x + 28$ .

$\frac{(x - 4) (x + 9) (x^{2} - 11 x + 28)}{(x + 7) (x - 7)}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $x^{2} - 11 x + 28$ using the AC method.

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Step 4.1.1

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 $28$ and whose sum is $- 11$ .

$- 7, - 4$

Step 4.1.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 9) ((x - 7) (x - 4))}{(x + 7) (x - 7)}$

$\frac{(x - 4) (x + 9) (x - 7) (x - 4)}{(x + 7) (x - 7)}$

Step 4.2

Combine exponents.

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Step 4.2.1

Raise $x - 4$ to the power of $1$ .

$\frac{{(x - 4)}^{1} (x - 4) (x + 9) (x - 7)}{(x + 7) (x - 7)}$

Step 4.2.2

Raise $x - 4$ to the power of $1$ .

$\frac{{(x - 4)}^{1} {(x - 4)}^{1} (x + 9) (x - 7)}{(x + 7) (x - 7)}$

Step 4.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(x - 4)}^{1 + 1} (x + 9) (x - 7)}{(x + 7) (x - 7)}$

Step 4.2.4

Add $1$ and $1$ .

$\frac{{(x - 4)}^{2} (x + 9) (x - 7)}{(x + 7) (x - 7)}$

Step 5

Cancel the common factor of $x - 7$ .

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Step 5.1

Cancel the common factor.

$\frac{{(x - 4)}^{2} (x + 9) (x - 7)}{(x + 7) (x - 7)}$

Step 5.2

Rewrite the expression.

$\frac{{(x - 4)}^{2} (x + 9)}{x + 7}$