Algebra Examples

$\frac{5 x}{4 x + 40} + \frac{x + 100}{6} = \frac{30}{x^{2} - 100}$

Step 1

Simplify $\frac{5 x}{4 x + 40} + \frac{x + 100}{6}$ .

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Step 1.1

Factor $4$ out of $4 x + 40$ .

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Step 1.1.1

Factor $4$ out of $4 x$ .

$\frac{5 x}{4 (x) + 40} + \frac{x + 100}{6} = \frac{30}{x^{2} - 100}$

Step 1.1.2

Factor $4$ out of $40$ .

$\frac{5 x}{4 x + 4 \cdot 10} + \frac{x + 100}{6} = \frac{30}{x^{2} - 100}$

Step 1.1.3

Factor $4$ out of $4 x + 4 \cdot 10$ .

$\frac{5 x}{4 (x + 10)} + \frac{x + 100}{6} = \frac{30}{x^{2} - 100}$

Step 1.2

To write $\frac{5 x}{4 (x + 10)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5 x}{4 (x + 10)} \cdot \frac{3}{3} + \frac{x + 100}{6} = \frac{30}{x^{2} - 100}$

Step 1.3

To write $\frac{x + 100}{6}$ as a fraction with a common denominator, multiply by $\frac{2 (x + 10)}{2 (x + 10)}$ .

$\frac{5 x}{4 (x + 10)} \cdot \frac{3}{3} + \frac{x + 100}{6} \cdot \frac{2 (x + 10)}{2 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.4

Write each expression with a common denominator of $12 (x + 10)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{5 x}{4 (x + 10)}$ by $\frac{3}{3}$ .

$\frac{5 x \cdot 3}{4 (x + 10) \cdot 3} + \frac{x + 100}{6} \cdot \frac{2 (x + 10)}{2 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.4.2

Multiply $3$ by $4$ .

$\frac{5 x \cdot 3}{12 (x + 10)} + \frac{x + 100}{6} \cdot \frac{2 (x + 10)}{2 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.4.3

Multiply $\frac{x + 100}{6}$ by $\frac{2 (x + 10)}{2 (x + 10)}$ .

$\frac{5 x \cdot 3}{12 (x + 10)} + \frac{(x + 100) (2 (x + 10))}{6 (2 (x + 10))} = \frac{30}{x^{2} - 100}$

Step 1.4.4

Multiply $2$ by $6$ .

$\frac{5 x \cdot 3}{12 (x + 10)} + \frac{(x + 100) (2 (x + 10))}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{5 x \cdot 3 + (x + 100) (2 (x + 10))}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $3$ by $5$ .

$\frac{15 x + (x + 100) \cdot 2 (x + 10)}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.2

Apply the distributive property.

$\frac{15 x + (x \cdot 2 + 100 \cdot 2) (x + 10)}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.3

Move $2$ to the left of $x$ .

$\frac{15 x + (2 \cdot x + 100 \cdot 2) (x + 10)}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.4

Multiply $100$ by $2$ .

$\frac{15 x + (2 \cdot x + 200) (x + 10)}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.5

Expand $(2 x + 200) (x + 10)$ using the FOIL Method.

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Step 1.6.5.1

Apply the distributive property.

$\frac{15 x + 2 x (x + 10) + 200 (x + 10)}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.5.2

Apply the distributive property.

$\frac{15 x + 2 x \cdot x + 2 x \cdot 10 + 200 (x + 10)}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.5.3

Apply the distributive property.

$\frac{15 x + 2 x \cdot x + 2 x \cdot 10 + 200 x + 200 \cdot 10}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.6

Simplify and combine like terms.

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Step 1.6.6.1

Simplify each term.

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Step 1.6.6.1.1

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

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Step 1.6.6.1.1.1

Move $x$ .

$\frac{15 x + 2 (x \cdot x) + 2 x \cdot 10 + 200 x + 200 \cdot 10}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.6.1.1.2

Multiply $x$ by $x$ .

$\frac{15 x + 2 x^{2} + 2 x \cdot 10 + 200 x + 200 \cdot 10}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.6.1.2

Multiply $10$ by $2$ .

$\frac{15 x + 2 x^{2} + 20 x + 200 x + 200 \cdot 10}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.6.1.3

Multiply $200$ by $10$ .

$\frac{15 x + 2 x^{2} + 20 x + 200 x + 2000}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.6.2

Add $20 x$ and $200 x$ .

$\frac{15 x + 2 x^{2} + 220 x + 2000}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 1.6.7

Add $15 x$ and $220 x$ .

$\frac{2 x^{2} + 235 x + 2000}{12 (x + 10)} = \frac{30}{x^{2} - 100}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $100$ as $10^{2}$ .

$\frac{2 x^{2} + 235 x + 2000}{12 (x + 10)} = \frac{30}{x^{2} - 10^{2}}$

Step 2.2

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 $b = 10$ .

$\frac{2 x^{2} + 235 x + 2000}{12 (x + 10)} = \frac{30}{(x + 10) (x - 10)}$

Step 3

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

$x \approx - 9.33086437, 10.07874588$

Step 4