Solve by Factoring (5x)/4-3/x=1/4

$\frac{5 x}{4} - \frac{3}{x} = \frac{1}{4}$

Step 1

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

$\frac{5 x}{4} - \frac{3}{x} - \frac{1}{4} = 0$

Step 2

To write $\frac{5 x}{4}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{5 x}{4} \cdot \frac{x}{x} - \frac{3}{x} - \frac{1}{4} = 0$

Step 3

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

$\frac{5 x}{4} \cdot \frac{x}{x} - \frac{3}{x} \cdot \frac{4}{4} - \frac{1}{4} = 0$

Step 4

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

Multiply $\frac{5 x}{4}$ by $\frac{x}{x}$ .

$\frac{5 x \cdot x}{4 x} - \frac{3}{x} \cdot \frac{4}{4} - \frac{1}{4} = 0$

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

$\frac{5 x \cdot x}{4 x} - \frac{3 \cdot 4}{x \cdot 4} - \frac{1}{4} = 0$

Reorder the factors of $x \cdot 4$ .

$\frac{5 x \cdot x}{4 x} - \frac{3 \cdot 4}{4 x} - \frac{1}{4} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{5 x \cdot x - 3 \cdot 4}{4 x} - \frac{1}{4} = 0$

Step 6

Multiply $- 3$ by $4$ .

$\frac{5 x \cdot x - 12}{4 x} - \frac{1}{4} = 0$

Step 7

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{5 (x \cdot x) - 12}{4 x} - \frac{1}{4} = 0$

Multiply $x$ by $x$ .

$\frac{5 x^{2} - 12}{4 x} - \frac{1}{4} = 0$

Step 8

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

$\frac{5 x^{2} - 12}{4 x} - \frac{1}{4} \cdot \frac{x}{x} = 0$

Step 9

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

$\frac{5 x^{2} - 12}{4 x} - \frac{x}{4 x} = 0$

Step 10

Combine the numerators over the common denominator.

$\frac{5 x^{2} - 12 - x}{4 x} = 0$

Step 11

Reorder terms.

$\frac{5 x^{2} - x - 12}{4 x} = 0$

Step 12

Set the numerator equal to zero.

$5 x^{2} - x - 12 = 0$

Step 13

Solve the equation for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 5$ , $b = - 1$ , and $c = - 12$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (5 \cdot - 12)}}{2 \cdot 5}$

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 12}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot - 12$ .

Multiply $- 4$ by $5$ .

$x = \frac{1 \pm \sqrt{1 - 20 \cdot - 12}}{2 \cdot 5}$

Multiply $- 20$ by $- 12$ .

$x = \frac{1 \pm \sqrt{1 + 240}}{2 \cdot 5}$

Add $1$ and $240$ .

$x = \frac{1 \pm \sqrt{241}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{1 \pm \sqrt{241}}{10}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 12}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot - 12$ .

Multiply $- 4$ by $5$ .

$x = \frac{1 \pm \sqrt{1 - 20 \cdot - 12}}{2 \cdot 5}$

Multiply $- 20$ by $- 12$ .

$x = \frac{1 \pm \sqrt{1 + 240}}{2 \cdot 5}$

Add $1$ and $240$ .

$x = \frac{1 \pm \sqrt{241}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{1 \pm \sqrt{241}}{10}$

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{241}}{10}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 5 \cdot - 12}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot - 12$ .

Multiply $- 4$ by $5$ .

$x = \frac{1 \pm \sqrt{1 - 20 \cdot - 12}}{2 \cdot 5}$

Multiply $- 20$ by $- 12$ .

$x = \frac{1 \pm \sqrt{1 + 240}}{2 \cdot 5}$

Add $1$ and $240$ .

$x = \frac{1 \pm \sqrt{241}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{1 \pm \sqrt{241}}{10}$

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{241}}{10}$

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{241}}{10}, \frac{1 - \sqrt{241}}{10}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{241}}{10}, \frac{1 - \sqrt{241}}{10}$

Decimal Form:

$x = 1.65241746 \dots, - 1.45241746 \dots$

Solve by Factoring (5x)/4-3/x=1/4

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