Algebra Examples

$\frac{3 x^{2}}{x^{2} - x - 30} - \frac{3 x + 5}{x - 6}$

Step 1

Factor $x^{2} - x - 30$ 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 $- 30$ and whose sum is $- 1$ .

$- 6, 5$

Step 1.2

Write the factored form using these integers.

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

Step 2

To write $- \frac{3 x + 5}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{3 x^{2}}{(x - 6) (x + 5)} - \frac{3 x + 5}{x - 6} \cdot \frac{x + 5}{x + 5}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{3 x + 5}{x - 6}$ by $\frac{x + 5}{x + 5}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{3 x^{2} + (- (3 x) - 1 \cdot 5) (x + 5)}{(x - 6) (x + 5)}$

Step 4.2

Multiply $3$ by $- 1$ .

$\frac{3 x^{2} + (- 3 x - 1 \cdot 5) (x + 5)}{(x - 6) (x + 5)}$

Step 4.3

Multiply $- 1$ by $5$ .

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

Step 4.4

Expand $(- 3 x - 5) (x + 5)$ using the FOIL Method.

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Step 4.4.1

Apply the distributive property.

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

Step 4.4.2

Apply the distributive property.

$\frac{3 x^{2} - 3 x \cdot x - 3 x \cdot 5 - 5 (x + 5)}{(x - 6) (x + 5)}$

Step 4.4.3

Apply the distributive property.

$\frac{3 x^{2} - 3 x \cdot x - 3 x \cdot 5 - 5 x - 5 \cdot 5}{(x - 6) (x + 5)}$

Step 4.5

Simplify and combine like terms.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

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

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Step 4.5.1.1.1

Move $x$ .

$\frac{3 x^{2} - 3 (x \cdot x) - 3 x \cdot 5 - 5 x - 5 \cdot 5}{(x - 6) (x + 5)}$

Step 4.5.1.1.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} - 3 x^{2} - 3 x \cdot 5 - 5 x - 5 \cdot 5}{(x - 6) (x + 5)}$

Step 4.5.1.2

Multiply $5$ by $- 3$ .

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

Step 4.5.1.3

Multiply $- 5$ by $5$ .

$\frac{3 x^{2} - 3 x^{2} - 15 x - 5 x - 25}{(x - 6) (x + 5)}$

Step 4.5.2

Subtract $5 x$ from $- 15 x$ .

$\frac{3 x^{2} - 3 x^{2} - 20 x - 25}{(x - 6) (x + 5)}$

Step 4.6

Subtract $3 x^{2}$ from $3 x^{2}$ .

$\frac{0 - 20 x - 25}{(x - 6) (x + 5)}$

Step 4.7

Subtract $20 x$ from $0$ .

$\frac{- 20 x - 25}{(x - 6) (x + 5)}$

Step 4.8

Factor $5$ out of $- 20 x - 25$ .

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Step 4.8.1

Factor $5$ out of $- 20 x$ .

$\frac{5 (- 4 x) - 25}{(x - 6) (x + 5)}$

Step 4.8.2

Factor $5$ out of $- 25$ .

$\frac{5 (- 4 x) + 5 (- 5)}{(x - 6) (x + 5)}$

Step 4.8.3

Factor $5$ out of $5 (- 4 x) + 5 (- 5)$ .

$\frac{5 (- 4 x - 5)}{(x - 6) (x + 5)}$

Step 5

Simplify with factoring out.

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Step 5.1

Factor $- 1$ out of $- 4 x$ .

$\frac{5 (- (4 x) - 5)}{(x - 6) (x + 5)}$

Step 5.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{5 (- (4 x) - 1 (5))}{(x - 6) (x + 5)}$

Step 5.3

Factor $- 1$ out of $- (4 x) - 1 (5)$ .

$\frac{5 (- (4 x + 5))}{(x - 6) (x + 5)}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Rewrite $- (4 x + 5)$ as $- 1 (4 x + 5)$ .

$\frac{5 (- 1 (4 x + 5))}{(x - 6) (x + 5)}$

Step 5.4.2

Move the negative in front of the fraction.

$- \frac{5 (4 x + 5)}{(x - 6) (x + 5)}$