Algebra Examples

$\frac{x^{2} - 14 x + 33}{x^{2} - 6 x - 55} - \frac{5}{3 x^{2} + 15 x}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - 14 x + 33$ using the AC method.

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Step 1.1.1

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 1.1.2

Write the factored form using these integers.

$\frac{(x - 11) (x - 3)}{x^{2} - 6 x - 55} - \frac{5}{3 x^{2} + 15 x}$

Step 1.2

Factor $x^{2} - 6 x - 55$ using the AC method.

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Step 1.2.1

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 $- 55$ and whose sum is $- 6$ .

$- 11, 5$

Step 1.2.2

Write the factored form using these integers.

$\frac{(x - 11) (x - 3)}{(x - 11) (x + 5)} - \frac{5}{3 x^{2} + 15 x}$

Step 1.3

Cancel the common factor of $x - 11$ .

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Step 1.3.1

Cancel the common factor.

$\frac{(x - 11) (x - 3)}{(x - 11) (x + 5)} - \frac{5}{3 x^{2} + 15 x}$

Step 1.3.2

Rewrite the expression.

$\frac{x - 3}{x + 5} - \frac{5}{3 x^{2} + 15 x}$

Step 1.4

Factor $3 x$ out of $3 x^{2} + 15 x$ .

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Step 1.4.1

Factor $3 x$ out of $3 x^{2}$ .

$\frac{x - 3}{x + 5} - \frac{5}{3 x (x) + 15 x}$

Step 1.4.2

Factor $3 x$ out of $15 x$ .

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

Step 1.4.3

Factor $3 x$ out of $3 x (x) + 3 x (5)$ .

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

Step 2

To write $\frac{x - 3}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{3 x}{3 x}$ .

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

Step 3

Write each expression with a common denominator of $(x + 5) \cdot 3 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

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

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

Step 3.2

Reorder the factors of $(x + 5) (3 x)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 5.3

Multiply $- 3$ by $3$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Multiply $x$ by $x$ by adding the exponents.

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Step 5.5.1

Move $x$ .

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

Step 5.5.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} - 9 x - 5}{3 x (x + 5)}$