Solve 20/x+x=11

$$\frac { 20 } { x } + x = 11$$

Solve for x

$$x = \frac{\sqrt{41} + 11}{2} \approx 8.701562119$$

$$x = \frac{11 - \sqrt{41}}{2} \approx 2.298437881$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$20+xx=11x$$

Multiply x and x to get x^{2}.

$$20+x^{2}=11x$$

Subtract 11x from both sides.

$$20+x^{2}-11x=0$$

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.

$$x^{2}-11x+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -11 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 20}}{2}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 20}}{2}$$

Multiply -4 times 20.

$$x=\frac{-\left(-11\right)±\sqrt{121-80}}{2}$$

Add 121 to -80.

$$x=\frac{-\left(-11\right)±\sqrt{41}}{2}$$

The opposite of -11 is 11.

$$x=\frac{11±\sqrt{41}}{2}$$

Now solve the equation x=\frac{11±\sqrt{41}}{2} when ± is plus. Add 11 to \sqrt{41}.

$$x=\frac{\sqrt{41}+11}{2}$$

Now solve the equation x=\frac{11±\sqrt{41}}{2} when ± is minus. Subtract \sqrt{41} from 11.

$$x=\frac{11-\sqrt{41}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{41}+11}{2} x=\frac{11-\sqrt{41}}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$20+xx=11x$$

Multiply x and x to get x^{2}.

$$20+x^{2}=11x$$

Subtract 11x from both sides.

$$20+x^{2}-11x=0$$

Subtract 20 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-11x=-20$$

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.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-20+\left(-\frac{11}{2}\right)^{2}$$

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

$$x^{2}-11x+\frac{121}{4}=-20+\frac{121}{4}$$

Add -20 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{41}{4}$$

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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{41}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{41}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{\sqrt{41}}{2} x-\frac{11}{2}=-\frac{\sqrt{41}}{2}$$

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

$$x=\frac{\sqrt{41}+11}{2} x=\frac{11-\sqrt{41}}{2}$$