Solve 2/3(x-1)=sqrt{x}

$$\frac { 2 } { 3 } ( x - 1 ) = \sqrt { x }$$

Solve for x

$$x=4$$

Solution Steps

Square both sides of the equation.

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

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

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

Multiply \frac{2}{3} and -1 to get -\frac{2}{3}.

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

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

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

Calculate \sqrt{x} to the power of 2 and get x.

$$\frac{4}{9}x^{2}-\frac{8}{9}x+\frac{4}{9}=x$$

Subtract x from both sides.

$$\frac{4}{9}x^{2}-\frac{8}{9}x+\frac{4}{9}-x=0$$

Combine -\frac{8}{9}x and -x to get -\frac{17}{9}x.

$$\frac{4}{9}x^{2}-\frac{17}{9}x+\frac{4}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{4}{9} for a, -\frac{17}{9} for b, and \frac{4}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{17}{9}\right)±\sqrt{\left(-\frac{17}{9}\right)^{2}-4\times \frac{4}{9}\times \frac{4}{9}}}{2\times \frac{4}{9}}$$

Square -\frac{17}{9} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{17}{9}\right)±\sqrt{\frac{289}{81}-4\times \frac{4}{9}\times \frac{4}{9}}}{2\times \frac{4}{9}}$$

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

$$x=\frac{-\left(-\frac{17}{9}\right)±\sqrt{\frac{289}{81}-\frac{16}{9}\times \frac{4}{9}}}{2\times \frac{4}{9}}$$

Multiply -\frac{16}{9} times \frac{4}{9} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{17}{9}\right)±\sqrt{\frac{289-64}{81}}}{2\times \frac{4}{9}}$$

Add \frac{289}{81} to -\frac{64}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{17}{9}\right)±\sqrt{\frac{25}{9}}}{2\times \frac{4}{9}}$$

Take the square root of \frac{25}{9}.

$$x=\frac{-\left(-\frac{17}{9}\right)±\frac{5}{3}}{2\times \frac{4}{9}}$$

The opposite of -\frac{17}{9} is \frac{17}{9}.

$$x=\frac{\frac{17}{9}±\frac{5}{3}}{2\times \frac{4}{9}}$$

Multiply 2 times \frac{4}{9}.

$$x=\frac{\frac{17}{9}±\frac{5}{3}}{\frac{8}{9}}$$

Now solve the equation x=\frac{\frac{17}{9}±\frac{5}{3}}{\frac{8}{9}} when ± is plus. Add \frac{17}{9} to \frac{5}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{32}{9}}{\frac{8}{9}}$$

Divide \frac{32}{9} by \frac{8}{9} by multiplying \frac{32}{9} by the reciprocal of \frac{8}{9}.

$$x=4$$

Now solve the equation x=\frac{\frac{17}{9}±\frac{5}{3}}{\frac{8}{9}} when ± is minus. Subtract \frac{5}{3} from \frac{17}{9} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{2}{9}}{\frac{8}{9}}$$

Divide \frac{2}{9} by \frac{8}{9} by multiplying \frac{2}{9} by the reciprocal of \frac{8}{9}.

$$x=\frac{1}{4}$$

The equation is now solved.

$$x=4 x=\frac{1}{4}$$

Substitute 4 for x in the equation \frac{2}{3}\left(x-1\right)=\sqrt{x}.

$$\frac{2}{3}\left(4-1\right)=\sqrt{4}$$

Simplify. The value x=4 satisfies the equation.

$$2=2$$

Substitute \frac{1}{4} for x in the equation \frac{2}{3}\left(x-1\right)=\sqrt{x}.

$$\frac{2}{3}\left(\frac{1}{4}-1\right)=\sqrt{\frac{1}{4}}$$

Simplify. The value x=\frac{1}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{1}{2}=\frac{1}{2}$$

Equation \frac{2\left(x-1\right)}{3}=\sqrt{x} has a unique solution.

$$x=4$$