Algebra Examples

$\frac{x}{2 x + 14} + \frac{x - 4}{6} = \frac{3}{x^{2} + 2 x - 35}$

Step 1

Simplify $\frac{x}{2 x + 14} + \frac{x - 4}{6}$ .

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Step 1.1

Factor $2$ out of $2 x + 14$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{x}{2 (x) + 14} + \frac{x - 4}{6} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.1.2

Factor $2$ out of $14$ .

$\frac{x}{2 x + 2 \cdot 7} + \frac{x - 4}{6} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.1.3

Factor $2$ out of $2 x + 2 \cdot 7$ .

$\frac{x}{2 (x + 7)} + \frac{x - 4}{6} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.2

To write $\frac{x}{2 (x + 7)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{2 (x + 7)} \cdot \frac{3}{3} + \frac{x - 4}{6} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.3

To write $\frac{x - 4}{6}$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$\frac{x}{2 (x + 7)} \cdot \frac{3}{3} + \frac{x - 4}{6} \cdot \frac{x + 7}{x + 7} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.4

Write each expression with a common denominator of $6 (x + 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{x}{2 (x + 7)}$ by $\frac{3}{3}$ .

$\frac{x \cdot 3}{2 (x + 7) \cdot 3} + \frac{x - 4}{6} \cdot \frac{x + 7}{x + 7} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.4.2

Multiply $3$ by $2$ .

$\frac{x \cdot 3}{6 (x + 7)} + \frac{x - 4}{6} \cdot \frac{x + 7}{x + 7} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.4.3

Multiply $\frac{x - 4}{6}$ by $\frac{x + 7}{x + 7}$ .

$\frac{x \cdot 3}{6 (x + 7)} + \frac{(x - 4) (x + 7)}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{x \cdot 3 + (x - 4) (x + 7)}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Move $3$ to the left of $x$ .

$\frac{3 \cdot x + (x - 4) (x + 7)}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.6.2

Expand $(x - 4) (x + 7)$ using the FOIL Method.

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Step 1.6.2.1

Apply the distributive property.

$\frac{3 x + x (x + 7) - 4 (x + 7)}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.6.2.2

Apply the distributive property.

$\frac{3 x + x \cdot x + x \cdot 7 - 4 (x + 7)}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.6.2.3

Apply the distributive property.

$\frac{3 x + x \cdot x + x \cdot 7 - 4 x - 4 \cdot 7}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.6.3

Simplify and combine like terms.

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Step 1.6.3.1

Simplify each term.

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Step 1.6.3.1.1

Multiply $x$ by $x$ .

$\frac{3 x + x^{2} + x \cdot 7 - 4 x - 4 \cdot 7}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.6.3.1.2

Move $7$ to the left of $x$ .

$\frac{3 x + x^{2} + 7 \cdot x - 4 x - 4 \cdot 7}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.6.3.1.3

Multiply $- 4$ by $7$ .

$\frac{3 x + x^{2} + 7 x - 4 x - 28}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.6.3.2

Subtract $4 x$ from $7 x$ .

$\frac{3 x + x^{2} + 3 x - 28}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 1.6.4

Add $3 x$ and $3 x$ .

$\frac{x^{2} + 6 x - 28}{6 (x + 7)} = \frac{3}{x^{2} + 2 x - 35}$

Step 2

Factor $x^{2} + 2 x - 35$ 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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Step 2.2

Write the factored form using these integers.

$\frac{x^{2} + 6 x - 28}{6 (x + 7)} = \frac{3}{(x - 5) (x + 7)}$

Step 3

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

$x \approx - 8.97595799, 2.46741577, 5.50854222$

Step 4