Solve by Factoring 4/(c+4)=c/(c+25)

$\frac{4}{c + 4} = \frac{c}{c + 25}$

Step 1

Subtract $\frac{c}{c + 25}$ from both sides of the equation.

$\frac{4}{c + 4} - \frac{c}{c + 25} = 0$

Step 2

Simplify $\frac{4}{c + 4} - \frac{c}{c + 25}$ .

To write $\frac{4}{c + 4}$ as a fraction with a common denominator, multiply by $\frac{c + 25}{c + 25}$ .

$\frac{4}{c + 4} \cdot \frac{c + 25}{c + 25} - \frac{c}{c + 25} = 0$

To write $- \frac{c}{c + 25}$ as a fraction with a common denominator, multiply by $\frac{c + 4}{c + 4}$ .

$\frac{4}{c + 4} \cdot \frac{c + 25}{c + 25} - \frac{c}{c + 25} \cdot \frac{c + 4}{c + 4} = 0$

Write each expression with a common denominator of $(c + 4) (c + 25)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4}{c + 4}$ by $\frac{c + 25}{c + 25}$ .

$\frac{4 (c + 25)}{(c + 4) (c + 25)} - \frac{c}{c + 25} \cdot \frac{c + 4}{c + 4} = 0$

Multiply $\frac{c}{c + 25}$ by $\frac{c + 4}{c + 4}$ .

$\frac{4 (c + 25)}{(c + 4) (c + 25)} - \frac{c (c + 4)}{(c + 25) (c + 4)} = 0$

Reorder the factors of $(c + 4) (c + 25)$ .

$\frac{4 (c + 25)}{(c + 25) (c + 4)} - \frac{c (c + 4)}{(c + 25) (c + 4)} = 0$

Combine the numerators over the common denominator.

$\frac{4 (c + 25) - c (c + 4)}{(c + 25) (c + 4)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{4 c + 4 \cdot 25 - c (c + 4)}{(c + 25) (c + 4)} = 0$

Multiply $4$ by $25$ .

$\frac{4 c + 100 - c (c + 4)}{(c + 25) (c + 4)} = 0$

Apply the distributive property.

$\frac{4 c + 100 - c \cdot c - c \cdot 4}{(c + 25) (c + 4)} = 0$

Multiply $c$ by $c$ by adding the exponents.

Move $c$ .

$\frac{4 c + 100 - (c \cdot c) - c \cdot 4}{(c + 25) (c + 4)} = 0$

Multiply $c$ by $c$ .

$\frac{4 c + 100 - c^{2} - c \cdot 4}{(c + 25) (c + 4)} = 0$

Multiply $4$ by $- 1$ .

$\frac{4 c + 100 - c^{2} - 4 c}{(c + 25) (c + 4)} = 0$

Subtract $4 c$ from $4 c$ .

$\frac{100 - c^{2} + 0}{(c + 25) (c + 4)} = 0$

Add $100 - c^{2}$ and $0$ .

$\frac{100 - c^{2}}{(c + 25) (c + 4)} = 0$

Rewrite $100 - c^{2}$ in a factored form.

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

$\frac{10^{2} - c^{2}}{(c + 25) (c + 4)} = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 10$ and $b = c$ .

$\frac{(10 + c) (10 - c)}{(c + 25) (c + 4)} = 0$

Step 3

Set the numerator equal to zero.

$(10 + c) (10 - c) = 0$

Step 4

Solve the equation for $c$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$10 + c = 0$

$10 - c = 0$

Set $10 + c$ equal to $0$ and solve for $c$ .

Set $10 + c$ equal to $0$ .

$10 + c = 0$

Subtract $10$ from both sides of the equation.

$c = - 10$

Set $10 - c$ equal to $0$ and solve for $c$ .

Set $10 - c$ equal to $0$ .

$10 - c = 0$

Solve $10 - c = 0$ for $c$ .

Subtract $10$ from both sides of the equation.

$- c = - 10$

Divide each term in $- c = - 10$ by $- 1$ and simplify.

Divide each term in $- c = - 10$ by $- 1$ .

$\frac{- c}{- 1} = \frac{- 10}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{c}{1} = \frac{- 10}{- 1}$

Divide $c$ by $1$ .

$c = \frac{- 10}{- 1}$

Simplify the right side.

Divide $- 10$ by $- 1$ .

$c = 10$

The final solution is all the values that make $(10 + c) (10 - c) = 0$ true.

$c = - 10, 10$

Solve by Factoring 4/(c+4)=c/(c+25)

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