Solve 1/2x^2+1/6=2/9

$$\frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 6 } = \frac { 2 } { 9 }$$

Solve for x

$$x=\frac{1}{3}\approx 0.333333333$$

$$x=-\frac{1}{3}\approx -0.333333333$$

Steps Using Difference of Squares

Subtract \frac{2}{9} from both sides.

$$\frac{1}{2}x^{2}+\frac{1}{6}-\frac{2}{9}=0$$

Subtract \frac{2}{9} from \frac{1}{6} to get -\frac{1}{18}.

$$\frac{1}{2}x^{2}-\frac{1}{18}=0$$

Multiply both sides by 18.

$$9x^{2}-1=0$$

Consider 9x^{2}-1. Rewrite 9x^{2}-1 as \left(3x\right)^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(3x-1\right)\left(3x+1\right)=0$$

To find equation solutions, solve 3x-1=0 and 3x+1=0.

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

Steps by Finding Square Root

Subtract \frac{1}{6} from both sides.

$$\frac{1}{2}x^{2}=\frac{2}{9}-\frac{1}{6}$$

Subtract \frac{1}{6} from \frac{2}{9} to get \frac{1}{18}.

$$\frac{1}{2}x^{2}=\frac{1}{18}$$

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$x^{2}=\frac{1}{18}\times 2$$

Multiply \frac{1}{18} and 2 to get \frac{1}{9}.

$$x^{2}=\frac{1}{9}$$

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Subtract \frac{2}{9} from both sides.

$$\frac{1}{2}x^{2}+\frac{1}{6}-\frac{2}{9}=0$$

Subtract \frac{2}{9} from \frac{1}{6} to get -\frac{1}{18}.

$$\frac{1}{2}x^{2}-\frac{1}{18}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 0 for b, and -\frac{1}{18} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{1}{2}\left(-\frac{1}{18}\right)}}{2\times \frac{1}{2}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{1}{2}\left(-\frac{1}{18}\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x=\frac{0±\sqrt{-2\left(-\frac{1}{18}\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -\frac{1}{18}.

$$x=\frac{0±\sqrt{\frac{1}{9}}}{2\times \frac{1}{2}}$$

Take the square root of \frac{1}{9}.

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

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

$$x=\frac{0±\frac{1}{3}}{1}$$

Now solve the equation x=\frac{0±\frac{1}{3}}{1} when ± is plus.

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{0±\frac{1}{3}}{1} when ± is minus.

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

The equation is now solved.

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