Solve x-1/x+1+3/x=1/4

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

Solve for x (complex solution)

$$x=\frac{-7+\sqrt{95}i}{6}\approx -1.166666667+1.624465724i$$

$$x=\frac{-\sqrt{95}i-7}{6}\approx -1.166666667-1.624465724i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+1\right), the least common multiple of x+1,x,4.

$$4x\left(x-1\right)+\left(4x+4\right)\times 3=x\left(x+1\right)$$

Use the distributive property to multiply 4x by x-1.

$$4x^{2}-4x+\left(4x+4\right)\times 3=x\left(x+1\right)$$

Use the distributive property to multiply 4x+4 by 3.

$$4x^{2}-4x+12x+12=x\left(x+1\right)$$

Combine -4x and 12x to get 8x.

$$4x^{2}+8x+12=x\left(x+1\right)$$

Use the distributive property to multiply x by x+1.

$$4x^{2}+8x+12=x^{2}+x$$

Subtract x^{2} from both sides.

$$4x^{2}+8x+12-x^{2}=x$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+8x+12=x$$

Subtract x from both sides.

$$3x^{2}+8x+12-x=0$$

Combine 8x and -x to get 7x.

$$3x^{2}+7x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 7 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 3\times 12}}{2\times 3}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 3\times 12}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-7±\sqrt{49-12\times 12}}{2\times 3}$$

Multiply -12 times 12.

$$x=\frac{-7±\sqrt{49-144}}{2\times 3}$$

Add 49 to -144.

$$x=\frac{-7±\sqrt{-95}}{2\times 3}$$

Take the square root of -95.

$$x=\frac{-7±\sqrt{95}i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-7±\sqrt{95}i}{6}$$

Now solve the equation x=\frac{-7±\sqrt{95}i}{6} when ± is plus. Add -7 to i\sqrt{95}.

$$x=\frac{-7+\sqrt{95}i}{6}$$

Now solve the equation x=\frac{-7±\sqrt{95}i}{6} when ± is minus. Subtract i\sqrt{95} from -7.

$$x=\frac{-\sqrt{95}i-7}{6}$$

The equation is now solved.

$$x=\frac{-7+\sqrt{95}i}{6} x=\frac{-\sqrt{95}i-7}{6}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+1\right), the least common multiple of x+1,x,4.

$$4x\left(x-1\right)+\left(4x+4\right)\times 3=x\left(x+1\right)$$

Use the distributive property to multiply 4x by x-1.

$$4x^{2}-4x+\left(4x+4\right)\times 3=x\left(x+1\right)$$

Use the distributive property to multiply 4x+4 by 3.

$$4x^{2}-4x+12x+12=x\left(x+1\right)$$

Combine -4x and 12x to get 8x.

$$4x^{2}+8x+12=x\left(x+1\right)$$

Use the distributive property to multiply x by x+1.

$$4x^{2}+8x+12=x^{2}+x$$

Subtract x^{2} from both sides.

$$4x^{2}+8x+12-x^{2}=x$$

Combine 4x^{2} and -x^{2} to get 3x^{2}.

$$3x^{2}+8x+12=x$$

Subtract x from both sides.

$$3x^{2}+8x+12-x=0$$

Combine 8x and -x to get 7x.

$$3x^{2}+7x+12=0$$

Subtract 12 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}+7x=-12$$

Divide both sides by 3.

$$\frac{3x^{2}+7x}{3}=-\frac{12}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{7}{3}x=-\frac{12}{3}$$

Divide -12 by 3.

$$x^{2}+\frac{7}{3}x=-4$$

Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=-4+\left(\frac{7}{6}\right)^{2}$$

Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=-4+\frac{49}{36}$$

Add -4 to \frac{49}{36}.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=-\frac{95}{36}$$

Factor x^{2}+\frac{7}{3}x+\frac{49}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{6}\right)^{2}=-\frac{95}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{-\frac{95}{36}}$$

Simplify.

$$x+\frac{7}{6}=\frac{\sqrt{95}i}{6} x+\frac{7}{6}=-\frac{\sqrt{95}i}{6}$$

Subtract \frac{7}{6} from both sides of the equation.

$$x=\frac{-7+\sqrt{95}i}{6} x=\frac{-\sqrt{95}i-7}{6}$$