Algebra Examples

$\frac{z}{z - 5} + \frac{4}{z + 5} = \frac{50}{z^{2} - 25}$

Step 1

Subtract $\frac{50}{z^{2} - 25}$ from both sides of the equation.

$\frac{z}{z - 5} + \frac{4}{z + 5} - \frac{50}{z^{2} - 25} = 0$

Step 2

Simplify $\frac{z}{z - 5} + \frac{4}{z + 5} - \frac{50}{z^{2} - 25}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

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

$\frac{z}{z - 5} + \frac{4}{z + 5} - \frac{50}{z^{2} - 5^{2}} = 0$

Step 2.1.2

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

$\frac{z}{z - 5} + \frac{4}{z + 5} - \frac{50}{(z + 5) (z - 5)} = 0$

Step 2.2

To write $\frac{z}{z - 5}$ as a fraction with a common denominator, multiply by $\frac{z + 5}{z + 5}$ .

$\frac{z}{z - 5} \cdot \frac{z + 5}{z + 5} + \frac{4}{z + 5} - \frac{50}{(z + 5) (z - 5)} = 0$

Step 2.3

To write $\frac{4}{z + 5}$ as a fraction with a common denominator, multiply by $\frac{z - 5}{z - 5}$ .

$\frac{z}{z - 5} \cdot \frac{z + 5}{z + 5} + \frac{4}{z + 5} \cdot \frac{z - 5}{z - 5} - \frac{50}{(z + 5) (z - 5)} = 0$

Step 2.4

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

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Step 2.4.1

Multiply $\frac{z}{z - 5}$ by $\frac{z + 5}{z + 5}$ .

$\frac{z (z + 5)}{(z - 5) (z + 5)} + \frac{4}{z + 5} \cdot \frac{z - 5}{z - 5} - \frac{50}{(z + 5) (z - 5)} = 0$

Step 2.4.2

Multiply $\frac{4}{z + 5}$ by $\frac{z - 5}{z - 5}$ .

$\frac{z (z + 5)}{(z - 5) (z + 5)} + \frac{4 (z - 5)}{(z + 5) (z - 5)} - \frac{50}{(z + 5) (z - 5)} = 0$

Step 2.4.3

Reorder the factors of $(z - 5) (z + 5)$ .

$\frac{z (z + 5)}{(z + 5) (z - 5)} + \frac{4 (z - 5)}{(z + 5) (z - 5)} - \frac{50}{(z + 5) (z - 5)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{z (z + 5) + 4 (z - 5)}{(z + 5) (z - 5)} - \frac{50}{(z + 5) (z - 5)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{z (z + 5) + 4 (z - 5) - 50}{(z + 5) (z - 5)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

$\frac{z \cdot z + z \cdot 5 + 4 (z - 5) - 50}{(z + 5) (z - 5)} = 0$

Step 2.7.2

Multiply $z$ by $z$ .

$\frac{z^{2} + z \cdot 5 + 4 (z - 5) - 50}{(z + 5) (z - 5)} = 0$

Step 2.7.3

Move $5$ to the left of $z$ .

$\frac{z^{2} + 5 \cdot z + 4 (z - 5) - 50}{(z + 5) (z - 5)} = 0$

Step 2.7.4

Apply the distributive property.

$\frac{z^{2} + 5 z + 4 z + 4 \cdot - 5 - 50}{(z + 5) (z - 5)} = 0$

Step 2.7.5

Multiply $4$ by $- 5$ .

$\frac{z^{2} + 5 z + 4 z - 20 - 50}{(z + 5) (z - 5)} = 0$

Step 2.8

Add $5 z$ and $4 z$ .

$\frac{z^{2} + 9 z - 20 - 50}{(z + 5) (z - 5)} = 0$

Step 2.9

Subtract $50$ from $- 20$ .

$\frac{z^{2} + 9 z - 70}{(z + 5) (z - 5)} = 0$

Step 2.10

Factor $z^{2} + 9 z - 70$ using the AC method.

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Step 2.10.1

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 $- 70$ and whose sum is $9$ .

$- 5, 14$

Step 2.10.2

Write the factored form using these integers.

$\frac{(z - 5) (z + 14)}{(z + 5) (z - 5)} = 0$

Step 2.11

Cancel the common factor of $z - 5$ .

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Step 2.11.1

Cancel the common factor.

$\frac{(z - 5) (z + 14)}{(z + 5) (z - 5)} = 0$

Step 2.11.2

Rewrite the expression.

$\frac{z + 14}{z + 5} = 0$

Step 3

Set the numerator equal to zero.

$z + 14 = 0$

Step 4

Subtract $14$ from both sides of the equation.

$z = - 14$