Solve 1=1/2+1/3u^2

$$1 = \frac { 1 } { 2 } + \frac { 1 } { 3 } u ^ { 2 }$$

Solve for u

$$u = \frac{\sqrt{6}}{2} \approx 1.224744871$$

$$u = -\frac{\sqrt{6}}{2} \approx -1.224744871$$

Steps by Finding Square Root

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

$$\frac{1}{2}+\frac{1}{3}u^{2}=1$$

Subtract \frac{1}{2} from both sides.

$$\frac{1}{3}u^{2}=1-\frac{1}{2}$$

Subtract \frac{1}{2} from 1 to get \frac{1}{2}.

$$\frac{1}{3}u^{2}=\frac{1}{2}$$

Multiply both sides by 3, the reciprocal of \frac{1}{3}.

$$u^{2}=\frac{1}{2}\times 3$$

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

$$u^{2}=\frac{3}{2}$$

Take the square root of both sides of the equation.

$$u=\frac{\sqrt{6}}{2} u=-\frac{\sqrt{6}}{2}$$

Steps Using the Quadratic Formula

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

$$\frac{1}{2}+\frac{1}{3}u^{2}=1$$

Subtract 1 from both sides.

$$\frac{1}{2}+\frac{1}{3}u^{2}-1=0$$

Subtract 1 from \frac{1}{2} to get -\frac{1}{2}.

$$-\frac{1}{2}+\frac{1}{3}u^{2}=0$$

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.

$$\frac{1}{3}u^{2}-\frac{1}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{3} for a, 0 for b, and -\frac{1}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$u=\frac{0±\sqrt{0^{2}-4\times \frac{1}{3}\left(-\frac{1}{2}\right)}}{2\times \frac{1}{3}}$$

Square 0.

$$u=\frac{0±\sqrt{-4\times \frac{1}{3}\left(-\frac{1}{2}\right)}}{2\times \frac{1}{3}}$$

Multiply -4 times \frac{1}{3}.

$$u=\frac{0±\sqrt{-\frac{4}{3}\left(-\frac{1}{2}\right)}}{2\times \frac{1}{3}}$$

Multiply -\frac{4}{3} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$u=\frac{0±\sqrt{\frac{2}{3}}}{2\times \frac{1}{3}}$$

Take the square root of \frac{2}{3}.

$$u=\frac{0±\frac{\sqrt{6}}{3}}{2\times \frac{1}{3}}$$

Multiply 2 times \frac{1}{3}.

$$u=\frac{0±\frac{\sqrt{6}}{3}}{\frac{2}{3}}$$

Now solve the equation u=\frac{0±\frac{\sqrt{6}}{3}}{\frac{2}{3}} when ± is plus.

$$u=\frac{\sqrt{6}}{2}$$

Now solve the equation u=\frac{0±\frac{\sqrt{6}}{3}}{\frac{2}{3}} when ± is minus.

$$u=-\frac{\sqrt{6}}{2}$$

The equation is now solved.

$$u=\frac{\sqrt{6}}{2} u=-\frac{\sqrt{6}}{2}$$