Solve z^2-1/4=0

$$z ^ { 2 } - \frac { 1 } { 4 } = 0$$

Solve for z

$$z=\frac{1}{2}=0.5$$

$$z=-\frac{1}{2}=-0.5$$

Steps Using Difference of Squares

Multiply both sides by 4.

$$4z^{2}-1=0$$

Consider 4z^{2}-1. Rewrite 4z^{2}-1 as \left(2z\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2z-1\right)\left(2z+1\right)=0$$

To find equation solutions, solve 2z-1=0 and 2z+1=0.

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

Steps by Finding Square Root

Add \frac{1}{4} to both sides. Anything plus zero gives itself.

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

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$z^{2}-\frac{1}{4}=0$$

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

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

Square 0.

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

Multiply -4 times -\frac{1}{4}.

$$z=\frac{0±\sqrt{1}}{2}$$

Take the square root of 1.

$$z=\frac{0±1}{2}$$

Now solve the equation z=\frac{0±1}{2} when ± is plus. Divide 1 by 2.

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

Now solve the equation z=\frac{0±1}{2} when ± is minus. Divide -1 by 2.

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

The equation is now solved.

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