Algebra Examples

$\frac{1}{x} - \frac{x}{64} = 0$

Step 1

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$\frac{1}{x} \cdot \frac{64}{64} - \frac{x}{64} = 0$

Step 2

To write $- \frac{x}{64}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{x} \cdot \frac{64}{64} - \frac{x}{64} \cdot \frac{x}{x} = 0$

Step 3

Write each expression with a common denominator of $x \cdot 64$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{x}$ by $\frac{64}{64}$ .

$\frac{64}{x \cdot 64} - \frac{x}{64} \cdot \frac{x}{x} = 0$

Step 3.2

Multiply $\frac{x}{64}$ by $\frac{x}{x}$ .

$\frac{64}{x \cdot 64} - \frac{x \cdot x}{64 x} = 0$

Step 3.3

Reorder the factors of $x \cdot 64$ .

$\frac{64}{64 x} - \frac{x \cdot x}{64 x} = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{64 - x \cdot x}{64 x} = 0$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $64$ as $8^{2}$ .

$\frac{8^{2} - x \cdot x}{64 x} = 0$

Step 5.2

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

$\frac{8^{2} - x^{2}}{64 x} = 0$

Step 5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = x$ .

$\frac{(8 + x) (8 - x)}{64 x} = 0$

Step 6

Set the numerator equal to zero.

$(8 + x) (8 - x) = 0$

Step 7

Solve the equation for $x$ .

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Step 7.1

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

$8 + x = 0$

$8 - x = 0$

Step 7.2

Set $8 + x$ equal to $0$ and solve for $x$ .

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Step 7.2.1

Set $8 + x$ equal to $0$ .

$8 + x = 0$

Step 7.2.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 7.3

Set $8 - x$ equal to $0$ and solve for $x$ .

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Step 7.3.1

Set $8 - x$ equal to $0$ .

$8 - x = 0$

Step 7.3.2

Solve $8 - x = 0$ for $x$ .

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Step 7.3.2.1

Subtract $8$ from both sides of the equation.

$- x = - 8$

Step 7.3.2.2

Divide each term in $- x = - 8$ by $- 1$ and simplify.

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Step 7.3.2.2.1

Divide each term in $- x = - 8$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 8}{- 1}$

Step 7.3.2.2.2

Simplify the left side.

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Step 7.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 8}{- 1}$

Step 7.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 8}{- 1}$

Step 7.3.2.2.3

Simplify the right side.

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Step 7.3.2.2.3.1

Divide $- 8$ by $- 1$ .

$x = 8$

Step 7.4

The final solution is all the values that make $(8 + x) (8 - x) = 0$ true.

$x = - 8, 8$