Solve 75-32z^2=0

$$75-32 { z }^{ 2 } =0$$

Solve for z

$$z = \frac{5 \sqrt{6}}{8} \approx 1.530931089$$

$$z = -\frac{5 \sqrt{6}}{8} \approx -1.530931089$$

Steps by Finding Square Root

Subtract 75 from both sides. Anything subtracted from zero gives its negation.

$$-32z^{2}=-75$$

Divide both sides by -32.

$$z^{2}=\frac{-75}{-32}$$

Fraction \frac{-75}{-32} can be simplified to \frac{75}{32} by removing the negative sign from both the numerator and the denominator.

$$z^{2}=\frac{75}{32}$$

Take the square root of both sides of the equation.

$$z=\frac{5\sqrt{6}}{8} z=-\frac{5\sqrt{6}}{8}$$

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.

$$-32z^{2}+75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -32 for a, 0 for b, and 75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{0±\sqrt{0^{2}-4\left(-32\right)\times 75}}{2\left(-32\right)}$$

Square 0.

$$z=\frac{0±\sqrt{-4\left(-32\right)\times 75}}{2\left(-32\right)}$$

Multiply -4 times -32.

$$z=\frac{0±\sqrt{128\times 75}}{2\left(-32\right)}$$

Multiply 128 times 75.

$$z=\frac{0±\sqrt{9600}}{2\left(-32\right)}$$

Take the square root of 9600.

$$z=\frac{0±40\sqrt{6}}{2\left(-32\right)}$$

Multiply 2 times -32.

$$z=\frac{0±40\sqrt{6}}{-64}$$

Now solve the equation z=\frac{0±40\sqrt{6}}{-64} when ± is plus.

$$z=-\frac{5\sqrt{6}}{8}$$

Now solve the equation z=\frac{0±40\sqrt{6}}{-64} when ± is minus.

$$z=\frac{5\sqrt{6}}{8}$$

The equation is now solved.

$$z=-\frac{5\sqrt{6}}{8} z=\frac{5\sqrt{6}}{8}$$