Algebra Examples

$\frac{x^{2} - x + 3}{x^{2} + 2 x - 8} - \frac{x^{2} - 3 x - 5}{x^{2} + 2 x - 8}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{x^{2} - x + 3 - (x^{2} - 3 x - 5)}{x^{2} + 2 x - 8}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$\frac{x^{2} - x + 3 - x^{2} - (- 3 x) - - 5}{x^{2} + 2 x - 8}$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Multiply $- 3$ by $- 1$ .

$\frac{x^{2} - x + 3 - x^{2} + 3 x - - 5}{x^{2} + 2 x - 8}$

Step 1.2.2.2

Multiply $- 1$ by $- 5$ .

$\frac{x^{2} - x + 3 - x^{2} + 3 x + 5}{x^{2} + 2 x - 8}$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine the opposite terms in $x^{2} - x + 3 - x^{2} + 3 x + 5$ .

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Step 1.3.1.1

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

$\frac{- x + 3 + 0 + 3 x + 5}{x^{2} + 2 x - 8}$

Step 1.3.1.2

Add $- x + 3$ and $0$ .

$\frac{- x + 3 + 3 x + 5}{x^{2} + 2 x - 8}$

Step 1.3.2

Add $- x$ and $3 x$ .

$\frac{2 x + 3 + 5}{x^{2} + 2 x - 8}$

Step 1.3.3

Add $3$ and $5$ .

$\frac{2 x + 8}{x^{2} + 2 x - 8}$

Step 1.3.4

Factor $2$ out of $2 x + 8$ .

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Step 1.3.4.1

Factor $2$ out of $2 x$ .

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

Step 1.3.4.2

Factor $2$ out of $8$ .

$\frac{2 x + 2 \cdot 4}{x^{2} + 2 x - 8}$

Step 1.3.4.3

Factor $2$ out of $2 x + 2 \cdot 4$ .

$\frac{2 (x + 4)}{x^{2} + 2 x - 8}$

Step 2

Factor $x^{2} + 2 x - 8$ 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 $- 8$ and whose sum is $2$ .

$- 2, 4$

Step 2.2

Write the factored form using these integers.

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

Step 3

Cancel the common factor of $x + 4$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$\frac{2}{x - 2}$