Solve x+1=2/x+1

$$x + 1 = \frac { 2 } { x + 1 }$$

Solve for x

$$x=-\left(\sqrt{2}+1\right)\approx -2.414213562$$

$$x=\sqrt{2}-1\approx 0.414213562$$

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 x+1.

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

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

$$x^{2}+x+x+1=2$$

Combine x and x to get 2x.

$$x^{2}+2x+1=2$$

Subtract 2 from both sides.

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

Subtract 2 from 1 to get -1.

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

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

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

Square 2.

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

Multiply -4 times -1.

$$x=\frac{-2±\sqrt{4+4}}{2}$$

Add 4 to 4.

$$x=\frac{-2±\sqrt{8}}{2}$$

Take the square root of 8.

$$x=\frac{-2±2\sqrt{2}}{2}$$

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

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

Divide -2+2\sqrt{2} by 2.

$$x=\sqrt{2}-1$$

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

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

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

$$x=-\sqrt{2}-1$$

The equation is now solved.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$

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 x+1.

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

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

$$x^{2}+x+x+1=2$$

Combine x and x to get 2x.

$$x^{2}+2x+1=2$$

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

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+1=\sqrt{2} x+1=-\sqrt{2}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{2}-1 x=-\sqrt{2}-1$$