Solve x-22=1/x-2

$$x - 22 = \frac { 1 } { x - 2 }$$

Solve for x

$$x=\sqrt{101}+12\approx 22.049875621$$

$$x=12-\sqrt{101}\approx 1.950124379$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

$$\left(x-2\right)x+\left(x-2\right)\left(-22\right)=1$$

Use the distributive property to multiply x-2 by x.

$$x^{2}-2x+\left(x-2\right)\left(-22\right)=1$$

Use the distributive property to multiply x-2 by -22.

$$x^{2}-2x-22x+44=1$$

Combine -2x and -22x to get -24x.

$$x^{2}-24x+44=1$$

Subtract 1 from both sides.

$$x^{2}-24x+44-1=0$$

Subtract 1 from 44 to get 43.

$$x^{2}-24x+43=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -24 for b, and 43 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 43}}{2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 43}}{2}$$

Multiply -4 times 43.

$$x=\frac{-\left(-24\right)±\sqrt{576-172}}{2}$$

Add 576 to -172.

$$x=\frac{-\left(-24\right)±\sqrt{404}}{2}$$

Take the square root of 404.

$$x=\frac{-\left(-24\right)±2\sqrt{101}}{2}$$

The opposite of -24 is 24.

$$x=\frac{24±2\sqrt{101}}{2}$$

Now solve the equation x=\frac{24±2\sqrt{101}}{2} when ± is plus. Add 24 to 2\sqrt{101}.

$$x=\frac{2\sqrt{101}+24}{2}$$

Divide 24+2\sqrt{101} by 2.

$$x=\sqrt{101}+12$$

Now solve the equation x=\frac{24±2\sqrt{101}}{2} when ± is minus. Subtract 2\sqrt{101} from 24.

$$x=\frac{24-2\sqrt{101}}{2}$$

Divide 24-2\sqrt{101} by 2.

$$x=12-\sqrt{101}$$

The equation is now solved.

$$x=\sqrt{101}+12 x=12-\sqrt{101}$$

Steps for Completing the Square

Variable x cannot be equal to 2 since division by zero is not defined. Multiply both sides of the equation by x-2.

$$\left(x-2\right)x+\left(x-2\right)\left(-22\right)=1$$

Use the distributive property to multiply x-2 by x.

$$x^{2}-2x+\left(x-2\right)\left(-22\right)=1$$

Use the distributive property to multiply x-2 by -22.

$$x^{2}-2x-22x+44=1$$

Combine -2x and -22x to get -24x.

$$x^{2}-24x+44=1$$

Subtract 44 from both sides.

$$x^{2}-24x=1-44$$

Subtract 44 from 1 to get -43.

$$x^{2}-24x=-43$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-24x+\left(-12\right)^{2}=-43+\left(-12\right)^{2}$$

Square -12.

$$x^{2}-24x+144=-43+144$$

Add -43 to 144.

$$x^{2}-24x+144=101$$

Factor x^{2}-24x+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-12\right)^{2}=101$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-12\right)^{2}}=\sqrt{101}$$

Simplify.

$$x-12=\sqrt{101} x-12=-\sqrt{101}$$

Add 12 to both sides of the equation.

$$x=\sqrt{101}+12 x=12-\sqrt{101}$$