Solve 3z^2+5=0

$$3 z ^ { 2 } + 5 = 0$$

Solve for z

$$z=-\frac{\sqrt{15}i}{3}\approx -0-1.290994449i$$

$$z=\frac{\sqrt{15}i}{3}\approx 1.290994449i$$

Steps by Finding Square Root

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$3z^{2}=-5$$

Divide both sides by 3.

$$z^{2}=-\frac{5}{3}$$

The equation is now solved.

$$z=\frac{\sqrt{15}i}{3} z=-\frac{\sqrt{15}i}{3}$$

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.

$$3z^{2}+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 0 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{0±\sqrt{0^{2}-4\times 3\times 5}}{2\times 3}$$

Square 0.

$$z=\frac{0±\sqrt{-4\times 3\times 5}}{2\times 3}$$

Multiply -4 times 3.

$$z=\frac{0±\sqrt{-12\times 5}}{2\times 3}$$

Multiply -12 times 5.

$$z=\frac{0±\sqrt{-60}}{2\times 3}$$

Take the square root of -60.

$$z=\frac{0±2\sqrt{15}i}{2\times 3}$$

Multiply 2 times 3.

$$z=\frac{0±2\sqrt{15}i}{6}$$

Now solve the equation z=\frac{0±2\sqrt{15}i}{6} when ± is plus.

$$z=\frac{\sqrt{15}i}{3}$$

Now solve the equation z=\frac{0±2\sqrt{15}i}{6} when ± is minus.

$$z=-\frac{\sqrt{15}i}{3}$$

The equation is now solved.

$$z=\frac{\sqrt{15}i}{3} z=-\frac{\sqrt{15}i}{3}$$