Algebra Examples

$\frac{2 x + 10}{x^{2} + x - 2} + \frac{2}{x + 2} + \frac{x}{x - 1}$

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $2 x + 10$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 10}{x^{2} + x - 2} + \frac{2}{x + 2} + \frac{x}{x - 1}$

Step 1.1.2

Factor $2$ out of $10$ .

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

Step 1.1.3

Factor $2$ out of $2 x + 2 \cdot 5$ .

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

Step 1.2

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

$- 1, 2$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

Find the common denominator.

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Step 2.1

Multiply $\frac{2}{x + 2}$ by $\frac{x - 1}{x - 1}$ .

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

Step 2.2

Multiply $\frac{2}{x + 2}$ by $\frac{x - 1}{x - 1}$ .

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

Step 2.3

Multiply $\frac{x}{x - 1}$ by $\frac{x + 2}{x + 2}$ .

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

Step 2.4

Multiply $\frac{x}{x - 1}$ by $\frac{x + 2}{x + 2}$ .

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

Step 3

Simplify terms.

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Step 3.1

Combine the numerators over the common denominator.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Multiply $2$ by $5$ .

$\frac{2 x + 10 + 2 (x - 1) + x (x + 2)}{(x + 2) (x - 1)}$

Step 3.2.3

Apply the distributive property.

$\frac{2 x + 10 + 2 x + 2 \cdot - 1 + x (x + 2)}{(x + 2) (x - 1)}$

Step 3.2.4

Multiply $2$ by $- 1$ .

$\frac{2 x + 10 + 2 x - 2 + x (x + 2)}{(x + 2) (x - 1)}$

Step 3.2.5

Apply the distributive property.

$\frac{2 x + 10 + 2 x - 2 + x \cdot x + x \cdot 2}{(x + 2) (x - 1)}$

Step 3.2.6

Multiply $x$ by $x$ .

$\frac{2 x + 10 + 2 x - 2 + x^{2} + x \cdot 2}{(x + 2) (x - 1)}$

Step 3.2.7

Move $2$ to the left of $x$ .

$\frac{2 x + 10 + 2 x - 2 + x^{2} + 2 x}{(x + 2) (x - 1)}$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Add $2 x$ and $2 x$ .

$\frac{4 x + 10 - 2 + x^{2} + 2 x}{(x + 2) (x - 1)}$

Step 3.3.2

Add $4 x$ and $2 x$ .

$\frac{6 x + 10 - 2 + x^{2}}{(x + 2) (x - 1)}$

Step 3.3.3

Simplify the expression.

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Step 3.3.3.1

Subtract $2$ from $10$ .

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

Step 3.3.3.2

Move $8$ .

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

Step 3.3.3.3

Reorder $6 x$ and $x^{2}$ .

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

Step 4

Factor $x^{2} + 6 x + 8$ using the AC method.

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Step 4.1

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 $6$ .

$2, 4$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 4)}{(x + 2) (x - 1)}$

Step 5

Cancel the common factor of $x + 2$ .

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Step 5.1

Cancel the common factor.

$\frac{(x + 2) (x + 4)}{(x + 2) (x - 1)}$

Step 5.2

Rewrite the expression.

$\frac{x + 4}{x - 1}$