Solve 1=132/x+11-132/x

$$1 = \frac { 132 } { x + 11 } - \frac { 132 } { x }$$

Solve for x (complex solution)

$$x=\frac{-11+11\sqrt{47}i}{2}\approx -5.5+37.706100302i$$

$$x=\frac{-11\sqrt{47}i-11}{2}\approx -5.5-37.706100302i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -11,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+11\right), the least common multiple of x+11,x.

$$x\left(x+11\right)=x\times 132-\left(x+11\right)\times 132$$

Use the distributive property to multiply x by x+11.

$$x^{2}+11x=x\times 132-\left(x+11\right)\times 132$$

Use the distributive property to multiply x+11 by 132.

$$x^{2}+11x=x\times 132-\left(132x+1452\right)$$

To find the opposite of 132x+1452, find the opposite of each term.

$$x^{2}+11x=x\times 132-132x-1452$$

Combine x\times 132 and -132x to get 0.

$$x^{2}+11x=-1452$$

Add 1452 to both sides.

$$x^{2}+11x+1452=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 11 for b, and 1452 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 1452}}{2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 1452}}{2}$$

Multiply -4 times 1452.

$$x=\frac{-11±\sqrt{121-5808}}{2}$$

Add 121 to -5808.

$$x=\frac{-11±\sqrt{-5687}}{2}$$

Take the square root of -5687.

$$x=\frac{-11±11\sqrt{47}i}{2}$$

Now solve the equation x=\frac{-11±11\sqrt{47}i}{2} when ± is plus. Add -11 to 11i\sqrt{47}.

$$x=\frac{-11+11\sqrt{47}i}{2}$$

Now solve the equation x=\frac{-11±11\sqrt{47}i}{2} when ± is minus. Subtract 11i\sqrt{47} from -11.

$$x=\frac{-11\sqrt{47}i-11}{2}$$

The equation is now solved.

$$x=\frac{-11+11\sqrt{47}i}{2} x=\frac{-11\sqrt{47}i-11}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -11,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+11\right), the least common multiple of x+11,x.

$$x\left(x+11\right)=x\times 132-\left(x+11\right)\times 132$$

Use the distributive property to multiply x by x+11.

$$x^{2}+11x=x\times 132-\left(x+11\right)\times 132$$

Use the distributive property to multiply x+11 by 132.

$$x^{2}+11x=x\times 132-\left(132x+1452\right)$$

To find the opposite of 132x+1452, find the opposite of each term.

$$x^{2}+11x=x\times 132-132x-1452$$

Combine x\times 132 and -132x to get 0.

$$x^{2}+11x=-1452$$

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.

$$x^{2}+11x+\left(\frac{11}{2}\right)^{2}=-1452+\left(\frac{11}{2}\right)^{2}$$

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

$$x^{2}+11x+\frac{121}{4}=-1452+\frac{121}{4}$$

Add -1452 to \frac{121}{4}.

$$x^{2}+11x+\frac{121}{4}=-\frac{5687}{4}$$

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}.

$$\left(x+\frac{11}{2}\right)^{2}=-\frac{5687}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{2}\right)^{2}}=\sqrt{-\frac{5687}{4}}$$

Simplify.

$$x+\frac{11}{2}=\frac{11\sqrt{47}i}{2} x+\frac{11}{2}=-\frac{11\sqrt{47}i}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$x=\frac{-11+11\sqrt{47}i}{2} x=\frac{-11\sqrt{47}i-11}{2}$$