Solve 1/2x^2-12=-37?

$$\frac { 1 } { 2 } x ^ { 2 } - 12 = - 37 ?$$

Solve for x (complex solution)

$$x=-5\sqrt{2}i\approx -0-7.071067812i$$

$$x=5\sqrt{2}i\approx 7.071067812i$$

Steps by Finding Square Root

Add 12 to both sides.

$$\frac{1}{2}x^{2}=-37+12$$

Add -37 and 12 to get -25.

$$\frac{1}{2}x^{2}=-25$$

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

$$x^{2}=-25\times 2$$

Multiply -25 and 2 to get -50.

$$x^{2}=-50$$

The equation is now solved.

$$x=5\sqrt{2}i x=-5\sqrt{2}i$$

Steps Using the Quadratic Formula

Add 37 to both sides.

$$\frac{1}{2}x^{2}-12+37=0$$

Add -12 and 37 to get 25.

$$\frac{1}{2}x^{2}+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 0 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{1}{2}\times 25}}{2\times \frac{1}{2}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{1}{2}\times 25}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x=\frac{0±\sqrt{-2\times 25}}{2\times \frac{1}{2}}$$

Multiply -2 times 25.

$$x=\frac{0±\sqrt{-50}}{2\times \frac{1}{2}}$$

Take the square root of -50.

$$x=\frac{0±5\sqrt{2}i}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{0±5\sqrt{2}i}{1}$$

Now solve the equation x=\frac{0±5\sqrt{2}i}{1} when ± is plus.

$$x=5\sqrt{2}i$$

Now solve the equation x=\frac{0±5\sqrt{2}i}{1} when ± is minus.

$$x=-5\sqrt{2}i$$

The equation is now solved.

$$x=5\sqrt{2}i x=-5\sqrt{2}i$$