Solve (x-1/3)^2+2/3x=10/9

$$( x - \frac { 1 } { 3 } ) ^ { 2 } + \frac { 2 } { 3 } x = \frac { 10 } { 9 }$$

Solve for x

$$x=-1$$

$$x=1$$

Steps Using Difference of Squares

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{1}{3}\right)^{2}.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}+\frac{2}{3}x=\frac{10}{9}$$

Combine -\frac{2}{3}x and \frac{2}{3}x to get 0.

$$x^{2}+\frac{1}{9}=\frac{10}{9}$$

Subtract \frac{10}{9} from both sides.

$$x^{2}+\frac{1}{9}-\frac{10}{9}=0$$

Subtract \frac{10}{9} from \frac{1}{9} to get -1.

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

Consider x^{2}-1. Rewrite x^{2}-1 as x^{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).

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

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

$$x=1 x=-1$$

Steps by Finding Square Root

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{1}{3}\right)^{2}.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}+\frac{2}{3}x=\frac{10}{9}$$

Combine -\frac{2}{3}x and \frac{2}{3}x to get 0.

$$x^{2}+\frac{1}{9}=\frac{10}{9}$$

Subtract \frac{1}{9} from both sides.

$$x^{2}=\frac{10}{9}-\frac{1}{9}$$

Subtract \frac{1}{9} from \frac{10}{9} to get 1.

$$x^{2}=1$$

Take the square root of both sides of the equation.

$$x=1 x=-1$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{1}{3}\right)^{2}.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}+\frac{2}{3}x=\frac{10}{9}$$

Combine -\frac{2}{3}x and \frac{2}{3}x to get 0.

$$x^{2}+\frac{1}{9}=\frac{10}{9}$$

Subtract \frac{10}{9} from both sides.

$$x^{2}+\frac{1}{9}-\frac{10}{9}=0$$

Subtract \frac{10}{9} from \frac{1}{9} to get -1.

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

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

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

Square 0.

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

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4}}{2}$$

Take the square root of 4.

$$x=\frac{0±2}{2}$$

Now solve the equation x=\frac{0±2}{2} when ± is plus. Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{0±2}{2} when ± is minus. Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=1 x=-1$$