Algebra Examples

$\frac{x^{2} + 5 x - 6}{x^{2} - 9 x + 8} \cdot \frac{x^{2}}{3 x + 18}$

Step 1

Factor $x^{2} + 5 x - 6$ 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 $- 6$ and whose sum is $5$ .

$- 1, 6$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 1) (x + 6)}{x^{2} - 9 x + 8} \cdot \frac{x^{2}}{3 x + 18}$

Step 2

Factor $x^{2} - 9 x + 8$ 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 $8$ and whose sum is $- 9$ .

$- 8, - 1$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 1) (x + 6)}{(x - 8) (x - 1)} \cdot \frac{x^{2}}{3 x + 18}$

Step 3

Simplify terms.

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Step 3.1

Factor $3$ out of $3 x + 18$ .

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Step 3.1.1

Factor $3$ out of $3 x$ .

$\frac{(x - 1) (x + 6)}{(x - 8) (x - 1)} \cdot \frac{x^{2}}{3 (x) + 18}$

Step 3.1.2

Factor $3$ out of $18$ .

$\frac{(x - 1) (x + 6)}{(x - 8) (x - 1)} \cdot \frac{x^{2}}{3 x + 3 \cdot 6}$

Step 3.1.3

Factor $3$ out of $3 x + 3 \cdot 6$ .

$\frac{(x - 1) (x + 6)}{(x - 8) (x - 1)} \cdot \frac{x^{2}}{3 (x + 6)}$

Step 3.2

Combine.

$\frac{(x - 1) (x + 6) x^{2}}{(x - 8) (x - 1) (3 (x + 6))}$

Step 3.3

Cancel the common factor of $x - 1$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(x - 1) (x + 6) x^{2}}{(x - 8) (x - 1) (3 (x + 6))}$

Step 3.3.2

Rewrite the expression.

$\frac{(x + 6) x^{2}}{(x - 8) (3 (x + 6))}$

Step 3.4

Cancel the common factor of $x + 6$ .

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Step 3.4.1

Cancel the common factor.

$\frac{(x + 6) x^{2}}{(x - 8) (3 (x + 6))}$

Step 3.4.2

Rewrite the expression.

$\frac{x^{2}}{(x - 8) \cdot (3)}$

Step 3.5

Move $3$ to the left of $x - 8$ .

$\frac{x^{2}}{3 (x - 8)}$