Algebra Examples

$\frac{x^{2} + x}{x^{2} - 4 x - 5} \div \frac{3 x^{2}}{x - 5}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + x}{x^{2} - 4 x - 5} \cdot \frac{x - 5}{3 x^{2}}$

Step 2

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

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Step 2.1

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

$\frac{x \cdot x + x}{x^{2} - 4 x - 5} \cdot \frac{x - 5}{3 x^{2}}$

Step 2.2

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

$\frac{x \cdot x + x^{1}}{x^{2} - 4 x - 5} \cdot \frac{x - 5}{3 x^{2}}$

Step 2.3

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot x + x \cdot 1}{x^{2} - 4 x - 5} \cdot \frac{x - 5}{3 x^{2}}$

Step 2.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$\frac{x (x + 1)}{x^{2} - 4 x - 5} \cdot \frac{x - 5}{3 x^{2}}$

Step 3

Factor $x^{2} - 4 x - 5$ using the AC method.

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Step 3.1

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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 5$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

$\frac{x + 1}{x + 1} \cdot \frac{1}{3 x}$

Step 4.3

Multiply $\frac{x + 1}{x + 1}$ by $\frac{1}{3 x}$ .

$\frac{x + 1}{(x + 1) (3 x)}$

Step 4.4

Cancel the common factor of $x + 1$ .

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Step 4.4.1

Cancel the common factor.

$\frac{x + 1}{(x + 1) (3 x)}$

Step 4.4.2

Rewrite the expression.

$\frac{1}{3 x}$