Solve x+33/12-x=x

$$\frac { x + 33 } { 12 - x } = x$$

Solve for x (complex solution)

$$x=\frac{11+\sqrt{11}i}{2}\approx 5.5+1.658312395i$$

$$x=\frac{-\sqrt{11}i+11}{2}\approx 5.5-1.658312395i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 12 since division by zero is not defined. Multiply both sides of the equation by -x+12.

$$x+33=x\left(-x+12\right)$$

Use the distributive property to multiply x by -x+12.

$$x+33=-x^{2}+12x$$

Add x^{2} to both sides.

$$x+33+x^{2}=12x$$

Subtract 12x from both sides.

$$x+33+x^{2}-12x=0$$

Combine x and -12x to get -11x.

$$-11x+33+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-11x+33=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -11 for b, and 33 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 33}}{2}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 33}}{2}$$

Multiply -4 times 33.

$$x=\frac{-\left(-11\right)±\sqrt{121-132}}{2}$$

Add 121 to -132.

$$x=\frac{-\left(-11\right)±\sqrt{-11}}{2}$$

Take the square root of -11.

$$x=\frac{-\left(-11\right)±\sqrt{11}i}{2}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{11}i}{2}$$

Now solve the equation x=\frac{11±\sqrt{11}i}{2} when ± is plus. Add 11 to i\sqrt{11}.

$$x=\frac{11+\sqrt{11}i}{2}$$

Now solve the equation x=\frac{11±\sqrt{11}i}{2} when ± is minus. Subtract i\sqrt{11} from 11.

$$x=\frac{-\sqrt{11}i+11}{2}$$

The equation is now solved.

$$x=\frac{11+\sqrt{11}i}{2} x=\frac{-\sqrt{11}i+11}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 12 since division by zero is not defined. Multiply both sides of the equation by -x+12.

$$x+33=x\left(-x+12\right)$$

Use the distributive property to multiply x by -x+12.

$$x+33=-x^{2}+12x$$

Add x^{2} to both sides.

$$x+33+x^{2}=12x$$

Subtract 12x from both sides.

$$x+33+x^{2}-12x=0$$

Combine x and -12x to get -11x.

$$-11x+33+x^{2}=0$$

Subtract 33 from both sides. Anything subtracted from zero gives its negation.

$$-11x+x^{2}=-33$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}-11x=-33$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-33+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=-33+\frac{121}{4}$$

Add -33 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=-\frac{11}{4}$$

Factor x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=-\frac{11}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{-\frac{11}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{\sqrt{11}i}{2} x-\frac{11}{2}=-\frac{\sqrt{11}i}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=\frac{11+\sqrt{11}i}{2} x=\frac{-\sqrt{11}i+11}{2}$$