Algebra Examples

$\frac{x}{x + 4} + \frac{12}{x^{2} + 5 x + 4} = \frac{8 x}{5 x - 15}$

Step 1

Simplify $\frac{x}{x + 4} + \frac{12}{x^{2} + 5 x + 4}$ .

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Step 1.1

Factor $x^{2} + 5 x + 4$ 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 $5$ .

$1, 4$

Step 1.1.2

Write the factored form using these integers.

$\frac{x}{x + 4} + \frac{12}{(x + 1) (x + 4)} = \frac{8 x}{5 x - 15}$

Step 1.2

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

$\frac{x}{x + 4} \cdot \frac{x + 1}{x + 1} + \frac{12}{(x + 1) (x + 4)} = \frac{8 x}{5 x - 15}$

Step 1.3

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

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Step 1.3.1

Multiply $\frac{x}{x + 4}$ by $\frac{x + 1}{x + 1}$ .

$\frac{x (x + 1)}{(x + 4) (x + 1)} + \frac{12}{(x + 1) (x + 4)} = \frac{8 x}{5 x - 15}$

Step 1.3.2

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

$\frac{x (x + 1)}{(x + 1) (x + 4)} + \frac{12}{(x + 1) (x + 4)} = \frac{8 x}{5 x - 15}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{x (x + 1) + 12}{(x + 1) (x + 4)} = \frac{8 x}{5 x - 15}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 1 + 12}{(x + 1) (x + 4)} = \frac{8 x}{5 x - 15}$

Step 1.5.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 1 + 12}{(x + 1) (x + 4)} = \frac{8 x}{5 x - 15}$

Step 1.5.3

Multiply $x$ by $1$ .

$\frac{x^{2} + x + 12}{(x + 1) (x + 4)} = \frac{8 x}{5 x - 15}$

Step 2

Factor $5$ out of $5 x - 15$ .

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Step 2.1

Factor $5$ out of $5 x$ .

$\frac{x^{2} + x + 12}{(x + 1) (x + 4)} = \frac{8 x}{5 (x) - 15}$

Step 2.2

Factor $5$ out of $- 15$ .

$\frac{x^{2} + x + 12}{(x + 1) (x + 4)} = \frac{8 x}{5 x + 5 \cdot - 3}$

Step 2.3

Factor $5$ out of $5 x + 5 \cdot - 3$ .

$\frac{x^{2} + x + 12}{(x + 1) (x + 4)} = \frac{8 x}{5 (x - 3)}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 17.12432675$

Step 4