Solve 1/3x^2+1-x^2=4/9

$$\frac { 1 } { 3 } x ^ { 2 } + 1 - x ^ { 2 } = \frac { 4 } { 9 }$$

Solve for x

$$x=\frac{\sqrt{30}}{6}\approx 0.912870929$$

$$x=-\frac{\sqrt{30}}{6}\approx -0.912870929$$

Steps by Finding Square Root

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

$$-\frac{2}{3}x^{2}+1=\frac{4}{9}$$

Subtract 1 from both sides.

$$-\frac{2}{3}x^{2}=\frac{4}{9}-1$$

Subtract 1 from \frac{4}{9} to get -\frac{5}{9}.

$$-\frac{2}{3}x^{2}=-\frac{5}{9}$$

Multiply both sides by -\frac{3}{2}, the reciprocal of -\frac{2}{3}.

$$x^{2}=-\frac{5}{9}\left(-\frac{3}{2}\right)$$

Multiply -\frac{5}{9} and -\frac{3}{2} to get \frac{5}{6}.

$$x^{2}=\frac{5}{6}$$

Take the square root of both sides of the equation.

$$x=\frac{\sqrt{30}}{6} x=-\frac{\sqrt{30}}{6}$$

Steps Using the Quadratic Formula

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

$$-\frac{2}{3}x^{2}+1=\frac{4}{9}$$

Subtract \frac{4}{9} from both sides.

$$-\frac{2}{3}x^{2}+1-\frac{4}{9}=0$$

Subtract \frac{4}{9} from 1 to get \frac{5}{9}.

$$-\frac{2}{3}x^{2}+\frac{5}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{2}{3} for a, 0 for b, and \frac{5}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-\frac{2}{3}\right)\times \frac{5}{9}}}{2\left(-\frac{2}{3}\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-\frac{2}{3}\right)\times \frac{5}{9}}}{2\left(-\frac{2}{3}\right)}$$

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

$$x=\frac{0±\sqrt{\frac{8}{3}\times \frac{5}{9}}}{2\left(-\frac{2}{3}\right)}$$

Multiply \frac{8}{3} times \frac{5}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0±\sqrt{\frac{40}{27}}}{2\left(-\frac{2}{3}\right)}$$

Take the square root of \frac{40}{27}.

$$x=\frac{0±\frac{2\sqrt{30}}{9}}{2\left(-\frac{2}{3}\right)}$$

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

$$x=\frac{0±\frac{2\sqrt{30}}{9}}{-\frac{4}{3}}$$

Now solve the equation x=\frac{0±\frac{2\sqrt{30}}{9}}{-\frac{4}{3}} when ± is plus.

$$x=-\frac{\sqrt{30}}{6}$$

Now solve the equation x=\frac{0±\frac{2\sqrt{30}}{9}}{-\frac{4}{3}} when ± is minus.

$$x=\frac{\sqrt{30}}{6}$$

The equation is now solved.

$$x=-\frac{\sqrt{30}}{6} x=\frac{\sqrt{30}}{6}$$