Solve by Factoring 1/x-x/16=0

$\frac{1}{x} - \frac{x}{16} = 0$

Step 1

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{1}{x} \cdot \frac{16}{16} - \frac{x}{16} = 0$

Step 2

To write $- \frac{x}{16}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{x} \cdot \frac{16}{16} - \frac{x}{16} \cdot \frac{x}{x} = 0$

Step 3

Write each expression with a common denominator of $x \cdot 16$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x}$ by $\frac{16}{16}$ .

$\frac{16}{x \cdot 16} - \frac{x}{16} \cdot \frac{x}{x} = 0$

Multiply $\frac{x}{16}$ by $\frac{x}{x}$ .

$\frac{16}{x \cdot 16} - \frac{x \cdot x}{16 x} = 0$

Reorder the factors of $x \cdot 16$ .

$\frac{16}{16 x} - \frac{x \cdot x}{16 x} = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{16 - x \cdot x}{16 x} = 0$

Step 5

Simplify the numerator.

Rewrite $16$ as $4^{2}$ .

$\frac{4^{2} - x \cdot x}{16 x} = 0$

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

$\frac{4^{2} - x^{2}}{16 x} = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = x$ .

$\frac{(4 + x) (4 - x)}{16 x} = 0$

Step 6

Set the numerator equal to zero.

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

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 $0$ .

$4 + x = 0$

$4 - x = 0$

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 1/x-x/16=0

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