Solve 1/3x^2-3=-x^2-2

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

Solve for x

$$x=\frac{\sqrt{3}}{2}\approx 0.866025404$$

$$x=-\frac{\sqrt{3}}{2}\approx -0.866025404$$

Steps by Finding Square Root

Add x^{2} to both sides.

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

Combine \frac{1}{3}x^{2} and x^{2} to get \frac{4}{3}x^{2}.

$$\frac{4}{3}x^{2}-3=-2$$

Add 3 to both sides.

$$\frac{4}{3}x^{2}=-2+3$$

Add -2 and 3 to get 1.

$$\frac{4}{3}x^{2}=1$$

Multiply both sides by \frac{3}{4}, the reciprocal of \frac{4}{3}.

$$x^{2}=1\times \frac{3}{4}$$

Multiply 1 and \frac{3}{4} to get \frac{3}{4}.

$$x^{2}=\frac{3}{4}$$

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Add x^{2} to both sides.

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

Combine \frac{1}{3}x^{2} and x^{2} to get \frac{4}{3}x^{2}.

$$\frac{4}{3}x^{2}-3=-2$$

Add 2 to both sides.

$$\frac{4}{3}x^{2}-3+2=0$$

Add -3 and 2 to get -1.

$$\frac{4}{3}x^{2}-1=0$$

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

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

Square 0.

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

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

$$x=\frac{0±\sqrt{-\frac{16}{3}\left(-1\right)}}{2\times \frac{4}{3}}$$

Multiply -\frac{16}{3} times -1.

$$x=\frac{0±\sqrt{\frac{16}{3}}}{2\times \frac{4}{3}}$$

Take the square root of \frac{16}{3}.

$$x=\frac{0±\frac{4\sqrt{3}}{3}}{2\times \frac{4}{3}}$$

Multiply 2 times \frac{4}{3}.

$$x=\frac{0±\frac{4\sqrt{3}}{3}}{\frac{8}{3}}$$

Now solve the equation x=\frac{0±\frac{4\sqrt{3}}{3}}{\frac{8}{3}} when ± is plus.

$$x=\frac{\sqrt{3}}{2}$$

Now solve the equation x=\frac{0±\frac{4\sqrt{3}}{3}}{\frac{8}{3}} when ± is minus.

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

The equation is now solved.

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