Solve 1/2x^2=9/32

$$\frac { 1 } { 2 } x ^ { 2 } = \frac { 9 } { 32 }$$

Solve for x

$$x=\frac{3}{4}=0.75$$

$$x=-\frac{3}{4}=-0.75$$

Steps Using Difference of Squares

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$x^{2}=\frac{9}{32}\times 2$$

Multiply \frac{9}{32} and 2 to get \frac{9}{16}.

$$x^{2}=\frac{9}{16}$$

Subtract \frac{9}{16} from both sides.

$$x^{2}-\frac{9}{16}=0$$

Multiply both sides by 16.

$$16x^{2}-9=0$$

Consider 16x^{2}-9. Rewrite 16x^{2}-9 as \left(4x\right)^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(4x-3\right)\left(4x+3\right)=0$$

To find equation solutions, solve 4x-3=0 and 4x+3=0.

$$x=\frac{3}{4} x=-\frac{3}{4}$$

Steps by Finding Square Root

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$x^{2}=\frac{9}{32}\times 2$$

Multiply \frac{9}{32} and 2 to get \frac{9}{16}.

$$x^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$x=\frac{3}{4} x=-\frac{3}{4}$$

Steps Using the Quadratic Formula

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$x^{2}=\frac{9}{32}\times 2$$

Multiply \frac{9}{32} and 2 to get \frac{9}{16}.

$$x^{2}=\frac{9}{16}$$

Subtract \frac{9}{16} from both sides.

$$x^{2}-\frac{9}{16}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -\frac{9}{16} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{9}{16}\right)}}{2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{9}{16}\right)}}{2}$$

Multiply -4 times -\frac{9}{16}.

$$x=\frac{0±\sqrt{\frac{9}{4}}}{2}$$

Take the square root of \frac{9}{4}.

$$x=\frac{0±\frac{3}{2}}{2}$$

Now solve the equation x=\frac{0±\frac{3}{2}}{2} when ± is plus.

$$x=\frac{3}{4}$$

Now solve the equation x=\frac{0±\frac{3}{2}}{2} when ± is minus.

$$x=-\frac{3}{4}$$

The equation is now solved.

$$x=\frac{3}{4} x=-\frac{3}{4}$$