Solve by Factoring 1/x+1/(x+3)=1/4

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

Step 1

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

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

Step 2

Simplify $\frac{1}{x} + \frac{1}{x + 3} - \frac{1}{4}$ .

Find the common denominator.

Multiply $\frac{1}{x}$ by $\frac{(x + 3) \cdot 4}{(x + 3) \cdot 4}$ .

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

Multiply $\frac{1}{x}$ by $\frac{(x + 3) \cdot 4}{(x + 3) \cdot 4}$ .

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

Multiply $\frac{1}{x + 3}$ by $\frac{x \cdot 4}{x \cdot 4}$ .

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

Multiply $\frac{1}{x + 3}$ by $\frac{x \cdot 4}{x \cdot 4}$ .

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

Multiply $\frac{1}{4}$ by $\frac{x (x + 3)}{x (x + 3)}$ .

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

Multiply $\frac{1}{4}$ by $\frac{x (x + 3)}{x (x + 3)}$ .

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

Reorder the factors of $x ((x + 3) \cdot 4)$ .

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

Reorder the factors of $(x + 3) (x \cdot 4)$ .

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

Reorder the factors of $4 (x (x + 3))$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Move $4$ to the left of $x$ .

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

Multiply $3$ by $4$ .

$\frac{4 \cdot x + 12 + x \cdot 4 - x (x + 3)}{4 x (x + 3)} = 0$

Move $4$ to the left of $x$ .

$\frac{4 x + 12 + 4 \cdot x - x (x + 3)}{4 x (x + 3)} = 0$

Apply the distributive property.

$\frac{4 x + 12 + 4 x - x \cdot x - x \cdot 3}{4 x (x + 3)} = 0$

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

Move $x$ .

$\frac{4 x + 12 + 4 x - (x \cdot x) - x \cdot 3}{4 x (x + 3)} = 0$

Multiply $x$ by $x$ .

$\frac{4 x + 12 + 4 x - x^{2} - x \cdot 3}{4 x (x + 3)} = 0$

Multiply $3$ by $- 1$ .

$\frac{4 x + 12 + 4 x - x^{2} - 3 x}{4 x (x + 3)} = 0$

Add $4 x$ and $4 x$ .

$\frac{8 x + 12 - x^{2} - 3 x}{4 x (x + 3)} = 0$

Subtract $3 x$ from $8 x$ .

$\frac{5 x + 12 - x^{2}}{4 x (x + 3)} = 0$

Reorder terms.

$\frac{- x^{2} + 5 x + 12}{4 x (x + 3)} = 0$

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) + 5 x + 12}{4 x (x + 3)} = 0$

Factor $- 1$ out of $5 x$ .

$\frac{- (x^{2}) - (- 5 x) + 12}{4 x (x + 3)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 5 x)$ .

$\frac{- (x^{2} - 5 x) + 12}{4 x (x + 3)} = 0$

Rewrite $12$ as $- 1 (- 12)$ .

$\frac{- (x^{2} - 5 x) - 1 (- 12)}{4 x (x + 3)} = 0$

Factor $- 1$ out of $- (x^{2} - 5 x) - 1 (- 12)$ .

$\frac{- (x^{2} - 5 x - 12)}{4 x (x + 3)} = 0$

Rewrite $- (x^{2} - 5 x - 12)$ as $- 1 (x^{2} - 5 x - 12)$ .

$\frac{- 1 (x^{2} - 5 x - 12)}{4 x (x + 3)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 5 x - 12}{4 x (x + 3)} = 0$

Step 3

Set the numerator equal to zero.

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

Step 4

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

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

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 12$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4$ by $- 12$ .

$x = \frac{5 \pm \sqrt{25 + 48}}{2 \cdot 1}$

Add $25$ and $48$ .

$x = \frac{5 \pm \sqrt{73}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{73}}{2}$

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

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 12$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4$ by $- 12$ .

$x = \frac{5 \pm \sqrt{25 + 48}}{2 \cdot 1}$

Add $25$ and $48$ .

$x = \frac{5 \pm \sqrt{73}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{73}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{73}}{2}$

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

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 12$ .

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 12}}{2 \cdot 1}$

Multiply $- 4$ by $- 12$ .

$x = \frac{5 \pm \sqrt{25 + 48}}{2 \cdot 1}$

Add $25$ and $48$ .

$x = \frac{5 \pm \sqrt{73}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{73}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{73}}{2}$

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{73}}{2}, \frac{5 - \sqrt{73}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{73}}{2}, \frac{5 - \sqrt{73}}{2}$

Decimal Form:

$x = 6.77200187 \dots, - 1.77200187 \dots$

Solve by Factoring 1/x+1/(x+3)=1/4

Related Questions

Support LCMGCF.com by disabling your adblocker.