Pre-Algebra Examples

$\frac{x}{x^{2} - 49} + \frac{3 x + 1}{x + 7}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $49$ as $7^{2}$ .

$\frac{x}{x^{2} - 7^{2}} + \frac{3 x + 1}{x + 7}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$\frac{x}{(x + 7) (x - 7)} + \frac{3 x + 1}{x + 7}$

Step 2

To write $\frac{3 x + 1}{x + 7}$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$\frac{x}{(x + 7) (x - 7)} + \frac{3 x + 1}{x + 7} \cdot \frac{x - 7}{x - 7}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{3 x + 1}{x + 7}$ by $\frac{x - 7}{x - 7}$ .

$\frac{x}{(x + 7) (x - 7)} + \frac{(3 x + 1) (x - 7)}{(x + 7) (x - 7)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{x + (3 x + 1) (x - 7)}{(x + 7) (x - 7)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(3 x + 1) (x - 7)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{x + 3 x (x - 7) + 1 (x - 7)}{(x + 7) (x - 7)}$

Step 4.1.2

Apply the distributive property.

$\frac{x + 3 x \cdot x + 3 x \cdot - 7 + 1 (x - 7)}{(x + 7) (x - 7)}$

Step 4.1.3

Apply the distributive property.

$\frac{x + 3 x \cdot x + 3 x \cdot - 7 + 1 x + 1 \cdot - 7}{(x + 7) (x - 7)}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Move $x$ .

$\frac{x + 3 (x \cdot x) + 3 x \cdot - 7 + 1 x + 1 \cdot - 7}{(x + 7) (x - 7)}$

Step 4.2.1.1.2

Multiply $x$ by $x$ .

$\frac{x + 3 x^{2} + 3 x \cdot - 7 + 1 x + 1 \cdot - 7}{(x + 7) (x - 7)}$

Step 4.2.1.2

Multiply $- 7$ by $3$ .

$\frac{x + 3 x^{2} - 21 x + 1 x + 1 \cdot - 7}{(x + 7) (x - 7)}$

Step 4.2.1.3

Multiply $x$ by $1$ .

$\frac{x + 3 x^{2} - 21 x + x + 1 \cdot - 7}{(x + 7) (x - 7)}$

Step 4.2.1.4

Multiply $- 7$ by $1$ .

$\frac{x + 3 x^{2} - 21 x + x - 7}{(x + 7) (x - 7)}$

Step 4.2.2

Add $- 21 x$ and $x$ .

$\frac{x + 3 x^{2} - 20 x - 7}{(x + 7) (x - 7)}$

Step 4.3

Subtract $20 x$ from $x$ .

$\frac{3 x^{2} - 19 x - 7}{(x + 7) (x - 7)}$