Solve by Factoring x/5-5/x=0

$\frac{x}{5} - \frac{5}{x} = 0$

Step 1

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

$\frac{x}{5} \cdot \frac{x}{x} - \frac{5}{x} = 0$

Step 2

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

$\frac{x}{5} \cdot \frac{x}{x} - \frac{5}{x} \cdot \frac{5}{5} = 0$

Step 3

Write each expression with a common denominator of $5 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x}{5}$ by $\frac{x}{x}$ .

$\frac{x \cdot x}{5 x} - \frac{5}{x} \cdot \frac{5}{5} = 0$

Multiply $\frac{5}{x}$ by $\frac{5}{5}$ .

$\frac{x \cdot x}{5 x} - \frac{5 \cdot 5}{x \cdot 5} = 0$

Reorder the factors of $x \cdot 5$ .

$\frac{x \cdot x}{5 x} - \frac{5 \cdot 5}{5 x} = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{x \cdot x - 5 \cdot 5}{5 x} = 0$

Step 5

Simplify the numerator.

Multiply $x$ by $x$ .

$\frac{x^{2} - 5 \cdot 5}{5 x} = 0$

Multiply $- 5$ by $5$ .

$\frac{x^{2} - 25}{5 x} = 0$

Rewrite $x^{2} - 25$ in a factored form.

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

$\frac{x^{2} - 5^{2}}{5 x} = 0$

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 $b = 5$ .

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

Step 6

Set the numerator equal to zero.

$(x + 5) (x - 5) = 0$

Step 7

Solve the equation for $x$ .

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

$x + 5 = 0$

$x - 5 = 0$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $(x + 5) (x - 5) = 0$ true.

$x = - 5, 5$

Solve by Factoring x/5-5/x=0

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