Solve by Factoring x/4-4/x=0

$\frac{x}{4} - \frac{4}{x} = 0$

Step 1

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

$\frac{x}{4} \cdot \frac{x}{x} - \frac{4}{x} = 0$

Step 2

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

$\frac{x}{4} \cdot \frac{x}{x} - \frac{4}{x} \cdot \frac{4}{4} = 0$

Step 3

Write each expression with a common denominator of $4 x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x}{4}$ by $\frac{x}{x}$ .

$\frac{x \cdot x}{4 x} - \frac{4}{x} \cdot \frac{4}{4} = 0$

Multiply $\frac{4}{x}$ by $\frac{4}{4}$ .

$\frac{x \cdot x}{4 x} - \frac{4 \cdot 4}{x \cdot 4} = 0$

Reorder the factors of $x \cdot 4$ .

$\frac{x \cdot x}{4 x} - \frac{4 \cdot 4}{4 x} = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{x \cdot x - 4 \cdot 4}{4 x} = 0$

Step 5

Simplify the numerator.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1} - 4 \cdot 4}{4 x} = 0$

Add $1$ and $1$ .

$\frac{x^{2} - 4 \cdot 4}{4 x} = 0$

Rewrite $4 \cdot 4$ as ${(2 \cdot 2)}^{2}$ .

$\frac{x^{2} - {(2 \cdot 2)}^{2}}{4 x} = 0$

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 $b = 2 \cdot 2$ .

$\frac{(x + 2 \cdot 2) (x - (2 \cdot 2))}{4 x} = 0$

Simplify.

Multiply $2$ by $2$ .

$\frac{(x + 4) (x - (2 \cdot 2))}{4 x} = 0$

Multiply $- (2 \cdot 2)$ .

Multiply $2$ by $2$ .

$\frac{(x + 4) (x - 1 \cdot 4)}{4 x} = 0$

Multiply $- 1$ by $4$ .

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

Step 6

Set the numerator equal to zero.

$(x + 4) (x - 4) = 0$

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 $0$ .

$x + 4 = 0$

$x - 4 = 0$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

The final solution is all the values that make $(x + 4) (x - 4) = 0$ true.

$x = - 4, 4$

Solve by Factoring x/4-4/x=0

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