Pre-Algebra Examples

$\frac{2 x}{x^{2} - 3 x - 4} + \frac{3}{5 x^{2} - 20 x}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} - 3 x - 4$ using the AC method.

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Step 1.1.1

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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 1.1.2

Write the factored form using these integers.

$\frac{2 x}{(x - 4) (x + 1)} + \frac{3}{5 x^{2} - 20 x}$

Step 1.2

Factor $5 x$ out of $5 x^{2} - 20 x$ .

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Step 1.2.1

Factor $5 x$ out of $5 x^{2}$ .

$\frac{2 x}{(x - 4) (x + 1)} + \frac{3}{5 x (x) - 20 x}$

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 2

To write $\frac{2 x}{(x - 4) (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{5 x}{5 x}$ .

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

Step 3

To write $\frac{3}{5 x (x - 4)}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 4

Write each expression with a common denominator of $(x - 4) (x + 1) \cdot 5 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2 x}{(x - 4) (x + 1)}$ by $\frac{5 x}{5 x}$ .

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

Step 4.2

Multiply $\frac{3}{5 x (x - 4)}$ by $\frac{x + 1}{x + 1}$ .

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

Step 4.3

Reorder the factors of $(x - 4) (x + 1) (5 x)$ .

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

Step 4.4

Reorder the factors of $5 x (x - 4) (x + 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Rewrite using the commutative property of multiplication.

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

Step 6.2

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

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Step 6.2.1

Move $x$ .

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

Step 6.2.2

Multiply $x$ by $x$ .

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

Step 6.3

Multiply $2$ by $5$ .

$\frac{10 x^{2} + 3 (x + 1)}{5 x (x + 1) (x - 4)}$

Step 6.4

Apply the distributive property.

$\frac{10 x^{2} + 3 x + 3 \cdot 1}{5 x (x + 1) (x - 4)}$

Step 6.5

Multiply $3$ by $1$ .

$\frac{10 x^{2} + 3 x + 3}{5 x (x + 1) (x - 4)}$