Solve x^2+2/3=x+(5/3)^2

$$x ^ { 2 } + \frac { 2 } { 3 } = x + ( \frac { 5 } { 3 } ) ^ { 2 }$$

Solve for x

$$x=\frac{\sqrt{85}}{6}+\frac{1}{2}\approx 2.036590743$$

$$x=-\frac{\sqrt{85}}{6}+\frac{1}{2}\approx -1.036590743$$

Steps Using the Quadratic Formula

Calculate \frac{5}{3} to the power of 2 and get \frac{25}{9}.

$$x^{2}+\frac{2}{3}=x+\frac{25}{9}$$

Subtract x from both sides.

$$x^{2}+\frac{2}{3}-x=\frac{25}{9}$$

Subtract \frac{25}{9} from both sides.

$$x^{2}+\frac{2}{3}-x-\frac{25}{9}=0$$

Subtract \frac{25}{9} from \frac{2}{3} to get -\frac{19}{9}.

$$x^{2}-\frac{19}{9}-x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-x-\frac{19}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -\frac{19}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-\frac{19}{9}\right)}}{2}$$

Multiply -4 times -\frac{19}{9}.

$$x=\frac{-\left(-1\right)±\sqrt{1+\frac{76}{9}}}{2}$$

Add 1 to \frac{76}{9}.

$$x=\frac{-\left(-1\right)±\sqrt{\frac{85}{9}}}{2}$$

Take the square root of \frac{85}{9}.

$$x=\frac{-\left(-1\right)±\frac{\sqrt{85}}{3}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\frac{\sqrt{85}}{3}}{2}$$

Now solve the equation x=\frac{1±\frac{\sqrt{85}}{3}}{2} when ± is plus. Add 1 to \frac{\sqrt{85}}{3}.

$$x=\frac{\frac{\sqrt{85}}{3}+1}{2}$$

Divide 1+\frac{\sqrt{85}}{3} by 2.

$$x=\frac{\sqrt{85}}{6}+\frac{1}{2}$$

Now solve the equation x=\frac{1±\frac{\sqrt{85}}{3}}{2} when ± is minus. Subtract \frac{\sqrt{85}}{3} from 1.

$$x=\frac{-\frac{\sqrt{85}}{3}+1}{2}$$

Divide 1-\frac{\sqrt{85}}{3} by 2.

$$x=-\frac{\sqrt{85}}{6}+\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{85}}{6}+\frac{1}{2} x=-\frac{\sqrt{85}}{6}+\frac{1}{2}$$

Steps for Completing the Square

Calculate \frac{5}{3} to the power of 2 and get \frac{25}{9}.

$$x^{2}+\frac{2}{3}=x+\frac{25}{9}$$

Subtract x from both sides.

$$x^{2}+\frac{2}{3}-x=\frac{25}{9}$$

Subtract \frac{2}{3} from both sides.

$$x^{2}-x=\frac{25}{9}-\frac{2}{3}$$

Subtract \frac{2}{3} from \frac{25}{9} to get \frac{19}{9}.

$$x^{2}-x=\frac{19}{9}$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{19}{9}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=\frac{19}{9}+\frac{1}{4}$$

Add \frac{19}{9} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{85}{36}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{85}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{85}{36}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{85}}{6} x-\frac{1}{2}=-\frac{\sqrt{85}}{6}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{85}}{6}+\frac{1}{2} x=-\frac{\sqrt{85}}{6}+\frac{1}{2}$$