Solve x=sqrt{8-2x/3}

$$x = \sqrt{ \frac{ 8-2x }{ 3 } }$$

Solve for x

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Solution Steps

Square both sides of the equation.

$$x^{2}=\left(\sqrt{\frac{8-2x}{3}}\right)^{2}$$

Calculate \sqrt{\frac{8-2x}{3}} to the power of 2 and get \frac{8-2x}{3}.

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

Divide each term of 8-2x by 3 to get \frac{8}{3}-\frac{2}{3}x.

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

Subtract \frac{8}{3} from both sides.

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

Add \frac{2}{3}x to both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{2}{3} for b, and -\frac{8}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{2}{3}±\sqrt{\left(\frac{2}{3}\right)^{2}-4\left(-\frac{8}{3}\right)}}{2}$$

Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{2}{3}±\sqrt{\frac{4}{9}-4\left(-\frac{8}{3}\right)}}{2}$$

Multiply -4 times -\frac{8}{3}.

$$x=\frac{-\frac{2}{3}±\sqrt{\frac{4}{9}+\frac{32}{3}}}{2}$$

Add \frac{4}{9} to \frac{32}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{2}{3}±\sqrt{\frac{100}{9}}}{2}$$

Take the square root of \frac{100}{9}.

$$x=\frac{-\frac{2}{3}±\frac{10}{3}}{2}$$

Now solve the equation x=\frac{-\frac{2}{3}±\frac{10}{3}}{2} when ± is plus. Add -\frac{2}{3} to \frac{10}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{8}{3}}{2}$$

Divide \frac{8}{3} by 2.

$$x=\frac{4}{3}$$

Now solve the equation x=\frac{-\frac{2}{3}±\frac{10}{3}}{2} when ± is minus. Subtract \frac{10}{3} from -\frac{2}{3} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=\frac{4}{3} x=-2$$

Substitute \frac{4}{3} for x in the equation x=\sqrt{\frac{8-2x}{3}}.

$$\frac{4}{3}=\sqrt{\frac{8-2\times \frac{4}{3}}{3}}$$

Simplify. The value x=\frac{4}{3} satisfies the equation.

$$\frac{4}{3}=\frac{4}{3}$$

Substitute -2 for x in the equation x=\sqrt{\frac{8-2x}{3}}.

$$-2=\sqrt{\frac{8-2\left(-2\right)}{3}}$$

Simplify. The value x=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=2$$

Equation x=\sqrt{\frac{8-2x}{3}} has a unique solution.

$$x=\frac{4}{3}$$