Solve x^2+2/3=x+5/8

$$x ^ { 2 } + \frac { 2 } { 3 } = x + \frac { 5 } { 8 }$$

Solve for x

$$x=\frac{\sqrt{30}}{12}+\frac{1}{2}\approx 0.956435465$$

$$x=-\frac{\sqrt{30}}{12}+\frac{1}{2}\approx 0.043564535$$

Steps Using the Quadratic Formula

Subtract x from both sides.

$$x^{2}+\frac{2}{3}-x=\frac{5}{8}$$

Subtract \frac{5}{8} from both sides.

$$x^{2}+\frac{2}{3}-x-\frac{5}{8}=0$$

Subtract \frac{5}{8} from \frac{2}{3} to get \frac{1}{24}.

$$x^{2}+\frac{1}{24}-x=0$$

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.

$$x^{2}-x+\frac{1}{24}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and \frac{1}{24} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times \frac{1}{24}}}{2}$$

Multiply -4 times \frac{1}{24}.

$$x=\frac{-\left(-1\right)±\sqrt{1-\frac{1}{6}}}{2}$$

Add 1 to -\frac{1}{6}.

$$x=\frac{-\left(-1\right)±\sqrt{\frac{5}{6}}}{2}$$

Take the square root of \frac{5}{6}.

$$x=\frac{-\left(-1\right)±\frac{\sqrt{30}}{6}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\frac{\sqrt{30}}{6}}{2}$$

Now solve the equation x=\frac{1±\frac{\sqrt{30}}{6}}{2} when ± is plus. Add 1 to \frac{\sqrt{30}}{6}.

$$x=\frac{\frac{\sqrt{30}}{6}+1}{2}$$

Divide 1+\frac{\sqrt{30}}{6} by 2.

$$x=\frac{\sqrt{30}}{12}+\frac{1}{2}$$

Now solve the equation x=\frac{1±\frac{\sqrt{30}}{6}}{2} when ± is minus. Subtract \frac{\sqrt{30}}{6} from 1.

$$x=\frac{-\frac{\sqrt{30}}{6}+1}{2}$$

Divide 1-\frac{\sqrt{30}}{6} by 2.

$$x=-\frac{\sqrt{30}}{12}+\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{30}}{12}+\frac{1}{2} x=-\frac{\sqrt{30}}{12}+\frac{1}{2}$$

Steps for Completing the Square

Subtract x from both sides.

$$x^{2}+\frac{2}{3}-x=\frac{5}{8}$$

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

$$x^{2}-x=\frac{5}{8}-\frac{2}{3}$$

Subtract \frac{2}{3} from \frac{5}{8} to get -\frac{1}{24}.

$$x^{2}-x=-\frac{1}{24}$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{1}{24}+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=-\frac{1}{24}+\frac{1}{4}$$

Add -\frac{1}{24} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{5}{24}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{5}{24}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{24}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{30}}{12} x-\frac{1}{2}=-\frac{\sqrt{30}}{12}$$

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

$$x=\frac{\sqrt{30}}{12}+\frac{1}{2} x=-\frac{\sqrt{30}}{12}+\frac{1}{2}$$