Solve x+1/x+2-1=2x

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

Solve for x

$$x=\frac{\sqrt{2}}{2}-1\approx -0.292893219$$

$$x=-\frac{\sqrt{2}}{2}-1\approx -1.707106781$$

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.

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

Use the distributive property to multiply x+2 by -1.

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

Combine x and -x to get 0.

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

Subtract 2 from 1 to get -1.

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

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

$$-1=2x^{2}+4x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}+4x=-1$$

Add 1 to both sides.

$$2x^{2}+4x+1=0$$

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

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8}}{2\times 2}$$

Add 16 to -8.

$$x=\frac{-4±\sqrt{8}}{2\times 2}$$

Take the square root of 8.

$$x=\frac{-4±2\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±2\sqrt{2}}{4}$$

Now solve the equation x=\frac{-4±2\sqrt{2}}{4} when ± is plus. Add -4 to 2\sqrt{2}.

$$x=\frac{2\sqrt{2}-4}{4}$$

Divide -4+2\sqrt{2} by 4.

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

Now solve the equation x=\frac{-4±2\sqrt{2}}{4} when ± is minus. Subtract 2\sqrt{2} from -4.

$$x=\frac{-2\sqrt{2}-4}{4}$$

Divide -4-2\sqrt{2} by 4.

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

The equation is now solved.

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

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.

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

Use the distributive property to multiply x+2 by -1.

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

Combine x and -x to get 0.

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

Subtract 2 from 1 to get -1.

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

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

$$-1=2x^{2}+4x$$

Swap sides so that all variable terms are on the left hand side.

$$2x^{2}+4x=-1$$

Divide both sides by 2.

$$\frac{2x^{2}+4x}{2}=-\frac{1}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{4}{2}x=-\frac{1}{2}$$

Divide 4 by 2.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 1.

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

Add -\frac{1}{2} to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract 1 from both sides of the equation.

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