Solve (z+1)(z-1)=1

$$( z + 1 ) ( z - 1 ) = 1$$

Solve for z

$$z=\sqrt{2}\approx 1.414213562$$

$$z=-\sqrt{2}\approx -1.414213562$$

Steps by Finding Square Root

Consider \left(z+1\right)\left(z-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Add 1 to both sides.

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

Add 1 and 1 to get 2.

$$z^{2}=2$$

Take the square root of both sides of the equation.

$$z=\sqrt{2} z=-\sqrt{2}$$

Steps Using the Quadratic Formula

Consider \left(z+1\right)\left(z-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

Subtract 1 from both sides.

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

Subtract 1 from -1 to get -2.

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

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

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

Square 0.

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

Multiply -4 times -2.

$$z=\frac{0±\sqrt{8}}{2}$$

Take the square root of 8.

$$z=\frac{0±2\sqrt{2}}{2}$$

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

$$z=\sqrt{2}$$

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

$$z=-\sqrt{2}$$

The equation is now solved.

$$z=\sqrt{2} z=-\sqrt{2}$$