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

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

Solve for v

$$v=-2\sqrt{6}i\approx -0-4.898979486i$$

$$v=2\sqrt{6}i\approx 4.898979486i$$

Steps by Finding Square Root

Calculate 3 to the power of 2 and get 9.

$$1=9+\frac{1}{3}v^{2}$$

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

$$9+\frac{1}{3}v^{2}=1$$

Subtract 9 from both sides.

$$\frac{1}{3}v^{2}=1-9$$

Subtract 9 from 1 to get -8.

$$\frac{1}{3}v^{2}=-8$$

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

$$v^{2}=-8\times 3$$

Multiply -8 and 3 to get -24.

$$v^{2}=-24$$

The equation is now solved.

$$v=2\sqrt{6}i v=-2\sqrt{6}i$$

Steps Using the Quadratic Formula

Calculate 3 to the power of 2 and get 9.

$$1=9+\frac{1}{3}v^{2}$$

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

$$9+\frac{1}{3}v^{2}=1$$

Subtract 1 from both sides.

$$9+\frac{1}{3}v^{2}-1=0$$

Subtract 1 from 9 to get 8.

$$8+\frac{1}{3}v^{2}=0$$

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.

$$\frac{1}{3}v^{2}+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{3} for a, 0 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{0±\sqrt{0^{2}-4\times \frac{1}{3}\times 8}}{2\times \frac{1}{3}}$$

Square 0.

$$v=\frac{0±\sqrt{-4\times \frac{1}{3}\times 8}}{2\times \frac{1}{3}}$$

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

$$v=\frac{0±\sqrt{-\frac{4}{3}\times 8}}{2\times \frac{1}{3}}$$

Multiply -\frac{4}{3} times 8.

$$v=\frac{0±\sqrt{-\frac{32}{3}}}{2\times \frac{1}{3}}$$

Take the square root of -\frac{32}{3}.

$$v=\frac{0±\frac{4\sqrt{6}i}{3}}{2\times \frac{1}{3}}$$

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

$$v=\frac{0±\frac{4\sqrt{6}i}{3}}{\frac{2}{3}}$$

Now solve the equation v=\frac{0±\frac{4\sqrt{6}i}{3}}{\frac{2}{3}} when ± is plus.

$$v=2\sqrt{6}i$$

Now solve the equation v=\frac{0±\frac{4\sqrt{6}i}{3}}{\frac{2}{3}} when ± is minus.

$$v=-2\sqrt{6}i$$

The equation is now solved.

$$v=2\sqrt{6}i v=-2\sqrt{6}i$$