Solve z-1=z^2+1

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

Solve for z

$$z=\frac{-\sqrt{7}i+1}{2}\approx 0.5-1.322875656i$$

$$z=\frac{1+\sqrt{7}i}{2}\approx 0.5+1.322875656i$$

Steps Using the Quadratic Formula

Subtract z^{2} from both sides.

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

Subtract 1 from both sides.

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

Subtract 1 from -1 to get -2.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

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

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

Square 1.

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

Multiply -4 times -1.

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

Multiply 4 times -2.

$$z=\frac{-1±\sqrt{1-8}}{2\left(-1\right)}$$

Add 1 to -8.

$$z=\frac{-1±\sqrt{-7}}{2\left(-1\right)}$$

Take the square root of -7.

$$z=\frac{-1±\sqrt{7}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$z=\frac{-1±\sqrt{7}i}{-2}$$

Now solve the equation z=\frac{-1±\sqrt{7}i}{-2} when ± is plus. Add -1 to i\sqrt{7}.

$$z=\frac{-1+\sqrt{7}i}{-2}$$

Divide -1+i\sqrt{7} by -2.

$$z=\frac{-\sqrt{7}i+1}{2}$$

Now solve the equation z=\frac{-1±\sqrt{7}i}{-2} when ± is minus. Subtract i\sqrt{7} from -1.

$$z=\frac{-\sqrt{7}i-1}{-2}$$

Divide -1-i\sqrt{7} by -2.

$$z=\frac{1+\sqrt{7}i}{2}$$

The equation is now solved.

$$z=\frac{-\sqrt{7}i+1}{2} z=\frac{1+\sqrt{7}i}{2}$$

Steps for Completing the Square

Subtract z^{2} from both sides.

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

Add 1 to both sides.

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

Add 1 and 1 to get 2.

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

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 1 by -1.

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

Divide 2 by -1.

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

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.

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

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

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

Add -2 to \frac{1}{4}.

$$z^{2}-z+\frac{1}{4}=-\frac{7}{4}$$

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}.

$$\left(z-\frac{1}{2}\right)^{2}=-\frac{7}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{7}{4}}$$

Simplify.

$$z-\frac{1}{2}=\frac{\sqrt{7}i}{2} z-\frac{1}{2}=-\frac{\sqrt{7}i}{2}$$

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

$$z=\frac{1+\sqrt{7}i}{2} z=\frac{-\sqrt{7}i+1}{2}$$