Solve for h 1/(h-5)+2/(h+5)=16/(h^2-25)

$\frac{1}{h - 5} + \frac{2}{h + 5} = \frac{16}{h^{2} - 25}$

Factor each term.

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Rewrite $25$ as $5^{2}$ .

$\frac{1}{h - 5} + \frac{2}{h + 5} = \frac{16}{h^{2} - 5^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = h$ and $b = 5$ .

$\frac{1}{h - 5} + \frac{2}{h + 5} = \frac{16}{(h + 5) (h - 5)}$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$h - 5, h + 5, (h + 5) (h - 5)$

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $h - 5$ is $h - 5$ itself.

$(h - 5) = h - 5$

$(h - 5)$ occurs $1$ time.

The factor for $h + 5$ is $h + 5$ itself.

$(h + 5) = h + 5$

$(h + 5)$ occurs $1$ time.

The factor for $h - 5$ is $h - 5$ itself.

$(h - 5) = h - 5$

$(h - 5)$ occurs $1$ time.

The LCM of $h - 5, h + 5, h + 5, h - 5$ is the result of multiplying all factors the greatest number of times they occur in either term.

$(h - 5) (h + 5)$

Multiply each term by $(h - 5) (h + 5)$ and simplify.

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Multiply each term in $\frac{1}{h - 5} + \frac{2}{h + 5} = \frac{16}{(h + 5) (h - 5)}$ by $(h - 5) (h + 5)$ in order to remove all the denominators from the equation.

$\frac{1}{h - 5} \cdot ((h - 5) (h + 5)) + \frac{2}{h + 5} \cdot ((h - 5) (h + 5)) = \frac{16}{(h + 5) (h - 5)} \cdot ((h - 5) (h + 5))$

Simplify $\frac{1}{h - 5} \cdot ((h - 5) (h + 5)) + \frac{2}{h + 5} \cdot ((h - 5) (h + 5))$ .

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Simplify each term.

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Cancel the common factor of $h - 5$ .

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Cancel the common factor.

$\frac{1}{h - 5} \cdot ((h - 5) (h + 5)) + \frac{2}{h + 5} \cdot ((h - 5) (h + 5)) = \frac{16}{(h + 5) (h - 5)} \cdot ((h - 5) (h + 5))$

Rewrite the expression.

$h + 5 + \frac{2}{h + 5} \cdot ((h - 5) (h + 5)) = \frac{16}{(h + 5) (h - 5)} \cdot ((h - 5) (h + 5))$

Cancel the common factor of $h + 5$ .

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Factor $h + 5$ out of $(h - 5) (h + 5)$ .

$h + 5 + \frac{2}{h + 5} \cdot ((h + 5) (h - 5)) = \frac{16}{(h + 5) (h - 5)} \cdot ((h - 5) (h + 5))$

Cancel the common factor.

$h + 5 + \frac{2}{h + 5} \cdot ((h + 5) (h - 5)) = \frac{16}{(h + 5) (h - 5)} \cdot ((h - 5) (h + 5))$

Rewrite the expression.

$h + 5 + 2 \cdot (h - 5) = \frac{16}{(h + 5) (h - 5)} \cdot ((h - 5) (h + 5))$

Apply the distributive property.

$h + 5 + 2 h + 2 \cdot - 5 = \frac{16}{(h + 5) (h - 5)} \cdot ((h - 5) (h + 5))$

Multiply $2$ by $- 5$ .

$h + 5 + 2 h - 10 = \frac{16}{(h + 5) (h - 5)} \cdot ((h - 5) (h + 5))$

Simplify by adding terms.

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Add $h$ and $2 h$ .

$3 h + 5 - 10 = \frac{16}{(h + 5) (h - 5)} \cdot ((h - 5) (h + 5))$

Subtract $10$ from $5$ .

$3 h - 5 = \frac{16}{(h + 5) (h - 5)} \cdot ((h - 5) (h + 5))$

Cancel the common factor of $(h - 5) (h + 5)$ .

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Factor $(h - 5) (h + 5)$ out of $(h + 5) (h - 5)$ .

$3 h - 5 = \frac{16}{(h - 5) (h + 5) (1)} \cdot ((h - 5) (h + 5))$

Cancel the common factor.

$3 h - 5 = \frac{16}{(h - 5) (h + 5) \cdot 1} \cdot ((h - 5) (h + 5))$

Rewrite the expression.

$3 h - 5 = 16$

Solve the equation.

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Move all terms not containing $h$ to the right side of the equation.

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Add $5$ to both sides of the equation.

$3 h = 16 + 5$

Add $16$ and $5$ .

$3 h = 21$

Divide each term by $3$ and simplify.

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Divide each term in $3 h = 21$ by $3$ .

$\frac{3 h}{3} = \frac{21}{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 h}{3} = \frac{21}{3}$

Divide $h$ by $1$ .

$h = \frac{21}{3}$

Divide $21$ by $3$ .

$h = 7$

Solve for h 1/(h-5)+2/(h+5)=16/(h^2-25)

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