Algebra Examples

$\frac{x}{4} - \frac{3}{x} = \frac{1}{4}$

Step 1

Subtract $\frac{1}{4}$ from both sides of the equation.

$\frac{x}{4} - \frac{3}{x} - \frac{1}{4} = 0$

Step 2

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

$\frac{x}{4} \cdot \frac{x}{x} - \frac{3}{x} - \frac{1}{4} = 0$

Step 3

To write $- \frac{3}{x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x}{4} \cdot \frac{x}{x} - \frac{3}{x} \cdot \frac{4}{4} - \frac{1}{4} = 0$

Step 4

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

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Step 4.1

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

$\frac{x \cdot x}{4 x} - \frac{3}{x} \cdot \frac{4}{4} - \frac{1}{4} = 0$

Step 4.2

Multiply $\frac{3}{x}$ by $\frac{4}{4}$ .

$\frac{x \cdot x}{4 x} - \frac{3 \cdot 4}{x \cdot 4} - \frac{1}{4} = 0$

Step 4.3

Reorder the factors of $x \cdot 4$ .

$\frac{x \cdot x}{4 x} - \frac{3 \cdot 4}{4 x} - \frac{1}{4} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{x \cdot x - 3 \cdot 4}{4 x} - \frac{1}{4} = 0$

Step 6

Multiply $- 3$ by $4$ .

$\frac{x \cdot x - 12}{4 x} - \frac{1}{4} = 0$

Step 7

Multiply $x$ by $x$ .

$\frac{x^{2} - 12}{4 x} - \frac{1}{4} = 0$

Step 8

To write $- \frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x^{2} - 12}{4 x} - \frac{1}{4} \cdot \frac{x}{x} = 0$

Step 9

Multiply $\frac{1}{4}$ by $\frac{x}{x}$ .

$\frac{x^{2} - 12}{4 x} - \frac{x}{4 x} = 0$

Step 10

Combine the numerators over the common denominator.

$\frac{x^{2} - 12 - x}{4 x} = 0$

Step 11

Factor $x^{2} - 12 - x$ using the AC method.

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Step 11.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $- 1$ .

$- 4, 3$

Step 11.2

Write the factored form using these integers.

$\frac{(x - 4) (x + 3)}{4 x} = 0$

Step 12

Set the numerator equal to zero.

$(x - 4) (x + 3) = 0$

Step 13

Solve the equation for $x$ .

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Step 13.1

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

$x - 4 = 0$

$x + 3 = 0$

Step 13.2

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

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Step 13.2.1

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

$x - 4 = 0$

Step 13.2.2

Add $4$ to both sides of the equation.

$x = 4$

Step 13.3

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 13.3.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 13.3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 13.4

The final solution is all the values that make $(x - 4) (x + 3) = 0$ true.

$x = 4, - 3$