Solve z^2=z+1

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

Solve for z

$$z = \frac{\sqrt{5} + 1}{2} \approx 1.618033989$$

$$z=\frac{1-\sqrt{5}}{2}\approx -0.618033989$$

Steps Using the Quadratic Formula

Subtract z from both sides.

$$z^{2}-z=1$$

Subtract 1 from both sides.

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

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

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

Multiply -4 times -1.

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

Add 1 to 4.

$$z=\frac{-\left(-1\right)±\sqrt{5}}{2}$$

The opposite of -1 is 1.

$$z=\frac{1±\sqrt{5}}{2}$$

Now solve the equation z=\frac{1±\sqrt{5}}{2} when ± is plus. Add 1 to \sqrt{5}.

$$z=\frac{\sqrt{5}+1}{2}$$

Now solve the equation z=\frac{1±\sqrt{5}}{2} when ± is minus. Subtract \sqrt{5} from 1.

$$z=\frac{1-\sqrt{5}}{2}$$

The equation is now solved.

$$z=\frac{\sqrt{5}+1}{2} z=\frac{1-\sqrt{5}}{2}$$

Steps for Completing the Square

Subtract z from both sides.

$$z^{2}-z=1$$

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

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add 1 to \frac{1}{4}.

$$z^{2}-z+\frac{1}{4}=\frac{5}{4}$$

Factor z^{2}-z+\frac{1}{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}.

$$\left(z-\frac{1}{2}\right)^{2}=\frac{5}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{4}}$$

Simplify.

$$z-\frac{1}{2}=\frac{\sqrt{5}}{2} z-\frac{1}{2}=-\frac{\sqrt{5}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$z=\frac{\sqrt{5}+1}{2} z=\frac{1-\sqrt{5}}{2}$$