Solve 1/2=frac{sqrt{2}}{6}-16x^2

$$\frac { 1 } { 2 } = \frac { \sqrt { 2 } } { 6 } - 16 x ^ { 2 }$$

Solve for x (complex solution)

$$x=-\frac{i\sqrt{18-6\sqrt{2}}}{24}\approx -0-0.12852474i$$

$$x=\frac{i\sqrt{18-6\sqrt{2}}}{24}\approx 0.12852474i$$

Steps by Finding Square Root

Multiply both sides of the equation by 6, the least common multiple of 2,6.

$$3=\sqrt{2}-96x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{2}-96x^{2}=3$$

Subtract \sqrt{2} from both sides.

$$-96x^{2}=3-\sqrt{2}$$

Dividing by -96 undoes the multiplication by -96.

$$x^{2}=\frac{3-\sqrt{2}}{-96}$$

Divide 3-\sqrt{2} by -96.

$$x^{2}=\frac{\sqrt{2}}{96}-\frac{1}{32}$$

Take the square root of both sides of the equation.

$$x=\frac{i\sqrt{18-6\sqrt{2}}}{24} x=-\frac{i\sqrt{18-6\sqrt{2}}}{24}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 2,6.

$$3=\sqrt{2}-96x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{2}-96x^{2}=3$$

Subtract 3 from both sides.

$$\sqrt{2}-96x^{2}-3=0$$

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.

$$-96x^{2}+\sqrt{2}-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -96 for a, 0 for b, and \sqrt{2}-3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-96\right)\left(\sqrt{2}-3\right)}}{2\left(-96\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-96\right)\left(\sqrt{2}-3\right)}}{2\left(-96\right)}$$

Multiply -4 times -96.

$$x=\frac{0±\sqrt{384\left(\sqrt{2}-3\right)}}{2\left(-96\right)}$$

Multiply 384 times \sqrt{2}-3.

$$x=\frac{0±\sqrt{384\sqrt{2}-1152}}{2\left(-96\right)}$$

Take the square root of 384\sqrt{2}-1152.

$$x=\frac{0±8i\sqrt{18-6\sqrt{2}}}{2\left(-96\right)}$$

Multiply 2 times -96.

$$x=\frac{0±8i\sqrt{18-6\sqrt{2}}}{-192}$$

Now solve the equation x=\frac{0±8i\sqrt{18-6\sqrt{2}}}{-192} when ± is plus.

$$x=-\frac{i\sqrt{18-6\sqrt{2}}}{24}$$

Now solve the equation x=\frac{0±8i\sqrt{18-6\sqrt{2}}}{-192} when ± is minus.

$$x=\frac{i\sqrt{18-6\sqrt{2}}}{24}$$

The equation is now solved.

$$x=-\frac{i\sqrt{18-6\sqrt{2}}}{24} x=\frac{i\sqrt{18-6\sqrt{2}}}{24}$$