Solve frac{2sqrt{x}+1}{3}=x

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

Solve for x

$$x=1$$

Solution Steps

Divide each term of 2\sqrt{x}+1 by 3 to get \frac{2}{3}\sqrt{x}+\frac{1}{3}.

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

Subtract x from both sides.

$$\frac{2}{3}\sqrt{x}+\frac{1}{3}-x=0$$

Subtract \frac{1}{3} from both sides. Anything subtracted from zero gives its negation.

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

Subtract -x from both sides of the equation.

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

Square both sides of the equation.

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

Expand \left(\frac{2}{3}\sqrt{x}\right)^{2}.

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

Calculate \frac{2}{3} to the power of 2 and get \frac{4}{9}.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-\frac{1}{3}+x\right)^{2}.

$$\frac{4}{9}x=\frac{1}{9}-\frac{2}{3}x+x^{2}$$

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

$$\frac{4}{9}x+\frac{2}{3}x=\frac{1}{9}+x^{2}$$

Combine \frac{4}{9}x and \frac{2}{3}x to get \frac{10}{9}x.

$$\frac{10}{9}x=\frac{1}{9}+x^{2}$$

Subtract x^{2} from both sides.

$$\frac{10}{9}x-x^{2}=\frac{1}{9}$$

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.

$$-x^{2}+\frac{10}{9}x=\frac{1}{9}$$

Subtract \frac{1}{9} from both sides of the equation.

$$-x^{2}+\frac{10}{9}x-\frac{1}{9}=\frac{1}{9}-\frac{1}{9}$$

Subtracting \frac{1}{9} from itself leaves 0.

$$-x^{2}+\frac{10}{9}x-\frac{1}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{10}{9} for b, and -\frac{1}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{10}{9}±\sqrt{\left(\frac{10}{9}\right)^{2}-4\left(-1\right)\left(-\frac{1}{9}\right)}}{2\left(-1\right)}$$

Square \frac{10}{9} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{10}{9}±\sqrt{\frac{100}{81}-4\left(-1\right)\left(-\frac{1}{9}\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\frac{10}{9}±\sqrt{\frac{100}{81}+4\left(-\frac{1}{9}\right)}}{2\left(-1\right)}$$

Multiply 4 times -\frac{1}{9}.

$$x=\frac{-\frac{10}{9}±\sqrt{\frac{100}{81}-\frac{4}{9}}}{2\left(-1\right)}$$

Add \frac{100}{81} to -\frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{10}{9}±\sqrt{\frac{64}{81}}}{2\left(-1\right)}$$

Take the square root of \frac{64}{81}.

$$x=\frac{-\frac{10}{9}±\frac{8}{9}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-\frac{10}{9}±\frac{8}{9}}{-2}$$

Now solve the equation x=\frac{-\frac{10}{9}±\frac{8}{9}}{-2} when ± is plus. Add -\frac{10}{9} to \frac{8}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -\frac{2}{9} by -2.

$$x=\frac{1}{9}$$

Now solve the equation x=\frac{-\frac{10}{9}±\frac{8}{9}}{-2} when ± is minus. Subtract \frac{8}{9} from -\frac{10}{9} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide -2 by -2.

$$x=1$$

The equation is now solved.

$$x=\frac{1}{9} x=1$$

Substitute \frac{1}{9} for x in the equation \frac{2\sqrt{x}+1}{3}=x.

$$\frac{2\sqrt{\frac{1}{9}}+1}{3}=\frac{1}{9}$$

Simplify. The value x=\frac{1}{9} does not satisfy the equation.

$$\frac{5}{9}=\frac{1}{9}$$

Substitute 1 for x in the equation \frac{2\sqrt{x}+1}{3}=x.

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

Simplify. The value x=1 satisfies the equation.

$$1=1$$

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

$$x=1$$