Basic Math Examples

$\frac{x + 5}{2 - x} \cdot \frac{x^{2} - 14 x + 33}{x^{2} - 13 x + 22}$

Step 1

Factor $x^{2} - 14 x + 33$ 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 $33$ and whose sum is $- 14$ .

$- 11, - 3$

Step 1.2

Write the factored form using these integers.

$\frac{x + 5}{2 - x} \cdot \frac{(x - 11) (x - 3)}{x^{2} - 13 x + 22}$

Step 2

Factor $x^{2} - 13 x + 22$ 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 $22$ and whose sum is $- 13$ .

$- 11, - 2$

Step 2.2

Write the factored form using these integers.

$\frac{x + 5}{2 - x} \cdot \frac{(x - 11) (x - 3)}{(x - 11) (x - 2)}$

Step 3

Cancel the common factor of $x - 11$ .

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Step 3.1

Cancel the common factor.

$\frac{x + 5}{2 - x} \cdot \frac{(x - 11) (x - 3)}{(x - 11) (x - 2)}$

Step 3.2

Rewrite the expression.

$\frac{x + 5}{2 - x} \cdot \frac{x - 3}{x - 2}$

Step 4

Multiply $\frac{x + 5}{2 - x} \cdot \frac{x - 3}{x - 2}$ .

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Step 4.1

Multiply $\frac{x + 5}{2 - x}$ by $\frac{x - 3}{x - 2}$ .

$\frac{(x + 5) (x - 3)}{(2 - x) (x - 2)}$

Step 4.2

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{(x + 5) (x - 3)}{(- 1 (- 2) - x) (x - 2)}$

Step 4.3

Factor $- 1$ out of $- x$ .

$\frac{(x + 5) (x - 3)}{(- 1 (- 2) - (x)) (x - 2)}$

Step 4.4

Factor $- 1$ out of $- 1 (- 2) - (x)$ .

$\frac{(x + 5) (x - 3)}{- 1 (- 2 + x) (x - 2)}$

Step 4.5

Reorder terms.

$\frac{(x + 5) (x - 3)}{- 1 (x - 2) (x - 2)}$

Step 4.6

Raise $x - 2$ to the power of $1$ .

$\frac{(x + 5) (x - 3)}{- 1 ({(x - 2)}^{1} (x - 2))}$

Step 4.7

Raise $x - 2$ to the power of $1$ .

$\frac{(x + 5) (x - 3)}{- 1 ({(x - 2)}^{1} {(x - 2)}^{1})}$

Step 4.8

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

$\frac{(x + 5) (x - 3)}{- 1 {(x - 2)}^{1 + 1}}$

Step 4.9

Add $1$ and $1$ .

$\frac{(x + 5) (x - 3)}{- 1 {(x - 2)}^{2}}$

Step 5

Move the negative in front of the fraction.

$- \frac{(x + 5) (x - 3)}{{(x - 2)}^{2}}$