Solve by Factoring 4/x=x/4

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

Step 1

Subtract $\frac{x}{4}$ from both sides of the equation.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

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

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

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

Reorder the factors of $x \cdot 4$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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

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

Rewrite $x \cdot x$ as $x^{2}$ .

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

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

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

Simplify.

Multiply $2$ by $2$ .

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

Multiply $2$ by $2$ .

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

Step 7

Set the numerator equal to zero.

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

Step 8

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 $0$ .

$4 + x = 0$

$4 - x = 0$

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

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

$4 + x = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

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

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

$4 - x = 0$

Solve $4 - x = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$- x = - 4$

Divide each term in $- x = - 4$ by $- 1$ and simplify.

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$x = 4$

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

$x = - 4, 4$

Solve by Factoring 4/x=x/4

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