Solve 2/1-x=x/4

$$\frac { 2 } { 1 - x } = \frac { x } { 4 }$$

Solve for x (complex solution)

$$x=\frac{1+\sqrt{31}i}{2}\approx 0.5+2.783882181i$$

$$x=\frac{-\sqrt{31}i+1}{2}\approx 0.5-2.783882181i$$

Steps Using the Quadratic Formula

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

$$-4\times 2=\left(x-1\right)x$$

Multiply -4 and 2 to get -8.

$$-8=\left(x-1\right)x$$

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

$$-8=x^{2}-x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-x=-8$$

Add 8 to both sides.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 8}}{2}$$

Multiply -4 times 8.

$$x=\frac{-\left(-1\right)±\sqrt{1-32}}{2}$$

Add 1 to -32.

$$x=\frac{-\left(-1\right)±\sqrt{-31}}{2}$$

Take the square root of -31.

$$x=\frac{-\left(-1\right)±\sqrt{31}i}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{31}i}{2}$$

Now solve the equation x=\frac{1±\sqrt{31}i}{2} when ± is plus. Add 1 to i\sqrt{31}.

$$x=\frac{1+\sqrt{31}i}{2}$$

Now solve the equation x=\frac{1±\sqrt{31}i}{2} when ± is minus. Subtract i\sqrt{31} from 1.

$$x=\frac{-\sqrt{31}i+1}{2}$$

The equation is now solved.

$$x=\frac{1+\sqrt{31}i}{2} x=\frac{-\sqrt{31}i+1}{2}$$

Steps for Completing the Square

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

$$-4\times 2=\left(x-1\right)x$$

Multiply -4 and 2 to get -8.

$$-8=\left(x-1\right)x$$

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

$$-8=x^{2}-x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-x=-8$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-8+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=-8+\frac{1}{4}$$

Add -8 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=-\frac{31}{4}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{31}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{31}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{31}i}{2} x-\frac{1}{2}=-\frac{\sqrt{31}i}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{1+\sqrt{31}i}{2} x=\frac{-\sqrt{31}i+1}{2}$$