Solve 2(x-1/3)^2+6=4-4/3x

$$2 ( x - \frac { 1 } { 3 } ) ^ { 2 } + 6 = 4 - \frac { 4 } { 3 } x$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{10}i}{3}\approx -0-1.054092553i$$

$$x=\frac{\sqrt{10}i}{3}\approx 1.054092553i$$

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}.

$$2\left(x^{2}-\frac{2}{3}x+\frac{1}{9}\right)+6=4-\frac{4}{3}x$$

Use the distributive property to multiply 2 by x^{2}-\frac{2}{3}x+\frac{1}{9}.

$$2x^{2}-\frac{4}{3}x+\frac{2}{9}+6=4-\frac{4}{3}x$$

Add \frac{2}{9} and 6 to get \frac{56}{9}.

$$2x^{2}-\frac{4}{3}x+\frac{56}{9}=4-\frac{4}{3}x$$

Add \frac{4}{3}x to both sides.

$$2x^{2}-\frac{4}{3}x+\frac{56}{9}+\frac{4}{3}x=4$$

Combine -\frac{4}{3}x and \frac{4}{3}x to get 0.

$$2x^{2}+\frac{56}{9}=4$$

Subtract \frac{56}{9} from both sides.

$$2x^{2}=4-\frac{56}{9}$$

Subtract \frac{56}{9} from 4 to get -\frac{20}{9}.

$$2x^{2}=-\frac{20}{9}$$

Divide both sides by 2.

$$x^{2}=\frac{-\frac{20}{9}}{2}$$

Express \frac{-\frac{20}{9}}{2} as a single fraction.

$$x^{2}=\frac{-20}{9\times 2}$$

Multiply 9 and 2 to get 18.

$$x^{2}=\frac{-20}{18}$$

Reduce the fraction \frac{-20}{18} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$x=\frac{\sqrt{10}i}{3} x=-\frac{\sqrt{10}i}{3}$$

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}.

$$2\left(x^{2}-\frac{2}{3}x+\frac{1}{9}\right)+6=4-\frac{4}{3}x$$

Use the distributive property to multiply 2 by x^{2}-\frac{2}{3}x+\frac{1}{9}.

$$2x^{2}-\frac{4}{3}x+\frac{2}{9}+6=4-\frac{4}{3}x$$

Add \frac{2}{9} and 6 to get \frac{56}{9}.

$$2x^{2}-\frac{4}{3}x+\frac{56}{9}=4-\frac{4}{3}x$$

Subtract 4 from both sides.

$$2x^{2}-\frac{4}{3}x+\frac{56}{9}-4=-\frac{4}{3}x$$

Subtract 4 from \frac{56}{9} to get \frac{20}{9}.

$$2x^{2}-\frac{4}{3}x+\frac{20}{9}=-\frac{4}{3}x$$

Add \frac{4}{3}x to both sides.

$$2x^{2}-\frac{4}{3}x+\frac{20}{9}+\frac{4}{3}x=0$$

Combine -\frac{4}{3}x and \frac{4}{3}x to get 0.

$$2x^{2}+\frac{20}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 0 for b, and \frac{20}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 2\times \frac{20}{9}}}{2\times 2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 2\times \frac{20}{9}}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{0±\sqrt{-8\times \frac{20}{9}}}{2\times 2}$$

Multiply -8 times \frac{20}{9}.

$$x=\frac{0±\sqrt{-\frac{160}{9}}}{2\times 2}$$

Take the square root of -\frac{160}{9}.

$$x=\frac{0±\frac{4\sqrt{10}i}{3}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{0±\frac{4\sqrt{10}i}{3}}{4}$$

Now solve the equation x=\frac{0±\frac{4\sqrt{10}i}{3}}{4} when ± is plus.

$$x=\frac{\sqrt{10}i}{3}$$

Now solve the equation x=\frac{0±\frac{4\sqrt{10}i}{3}}{4} when ± is minus.

$$x=-\frac{\sqrt{10}i}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{10}i}{3} x=-\frac{\sqrt{10}i}{3}$$