Solve 2x_{0}=x_0^2+1/x_{0-1}

$$2 x _ { 0 } = \frac { x _ { 0 } ^ { 2 } + 1 } { x _ { 0 } - 1 }$$

Solve for x_0

$$x_{0}=\sqrt{2}+1\approx 2.414213562$$

$$x_{0}=1-\sqrt{2}\approx -0.414213562$$

Steps Using the Quadratic Formula

Variable x_{0} cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x_{0}-1.

$$2x_{0}\left(x_{0}-1\right)=x_{0}^{2}+1$$

Use the distributive property to multiply 2x_{0} by x_{0}-1.

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

Subtract x_{0}^{2} from both sides.

$$2x_{0}^{2}-2x_{0}-x_{0}^{2}=1$$

Combine 2x_{0}^{2} and -x_{0}^{2} to get x_{0}^{2}.

$$x_{0}^{2}-2x_{0}=1$$

Subtract 1 from both sides.

$$x_{0}^{2}-2x_{0}-1=0$$

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}.

$$x_{0}=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)}}{2}$$

Square -2.

$$x_{0}=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$x_{0}=\frac{-\left(-2\right)±\sqrt{4+4}}{2}$$

Add 4 to 4.

$$x_{0}=\frac{-\left(-2\right)±\sqrt{8}}{2}$$

Take the square root of 8.

$$x_{0}=\frac{-\left(-2\right)±2\sqrt{2}}{2}$$

The opposite of -2 is 2.

$$x_{0}=\frac{2±2\sqrt{2}}{2}$$

Now solve the equation x_{0}=\frac{2±2\sqrt{2}}{2} when ± is plus. Add 2 to 2\sqrt{2}.

$$x_{0}=\frac{2\sqrt{2}+2}{2}$$

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

$$x_{0}=\sqrt{2}+1$$

Now solve the equation x_{0}=\frac{2±2\sqrt{2}}{2} when ± is minus. Subtract 2\sqrt{2} from 2.

$$x_{0}=\frac{2-2\sqrt{2}}{2}$$

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

$$x_{0}=1-\sqrt{2}$$

The equation is now solved.

$$x_{0}=\sqrt{2}+1 x_{0}=1-\sqrt{2}$$

Steps for Completing the Square

Variable x_{0} cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x_{0}-1.

$$2x_{0}\left(x_{0}-1\right)=x_{0}^{2}+1$$

Use the distributive property to multiply 2x_{0} by x_{0}-1.

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

Subtract x_{0}^{2} from both sides.

$$2x_{0}^{2}-2x_{0}-x_{0}^{2}=1$$

Combine 2x_{0}^{2} and -x_{0}^{2} to get x_{0}^{2}.

$$x_{0}^{2}-2x_{0}=1$$

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.

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

Add 1 to 1.

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

Factor x_{0}^{2}-2x_{0}+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}.

$$\left(x_{0}-1\right)^{2}=2$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x_{0}-1\right)^{2}}=\sqrt{2}$$

Simplify.

$$x_{0}-1=\sqrt{2} x_{0}-1=-\sqrt{2}$$

Add 1 to both sides of the equation.

$$x_{0}=\sqrt{2}+1 x_{0}=1-\sqrt{2}$$