Solve sqrt{x+1/3}+5x=3x-1

$$\sqrt { \frac { x + 1 } { 3 } } + 5 x = 3 x - 1$$

Solve for x

$$x=-\frac{2}{3}\approx -0.666666667$$

Solution Steps

Subtract 5x from both sides of the equation.

$$\sqrt{\frac{x+1}{3}}=3x-1-5x$$

Combine 3x and -5x to get -2x.

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

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

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

Square both sides of the equation.

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

Calculate \sqrt{\frac{1}{3}x+\frac{1}{3}} to the power of 2 and get \frac{1}{3}x+\frac{1}{3}.

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

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

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

Subtract 4x^{2} from both sides.

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

Subtract 4x from both sides.

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

Combine \frac{1}{3}x and -4x to get -\frac{11}{3}x.

$$-\frac{11}{3}x+\frac{1}{3}-4x^{2}=1$$

Subtract 1 from both sides.

$$-\frac{11}{3}x+\frac{1}{3}-4x^{2}-1=0$$

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

$$-\frac{11}{3}x-\frac{2}{3}-4x^{2}=0$$

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.

$$-4x^{2}-\frac{11}{3}x-\frac{2}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -\frac{11}{3} for b, and -\frac{2}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{11}{3}\right)±\sqrt{\left(-\frac{11}{3}\right)^{2}-4\left(-4\right)\left(-\frac{2}{3}\right)}}{2\left(-4\right)}$$

Square -\frac{11}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{11}{3}\right)±\sqrt{\frac{121}{9}-4\left(-4\right)\left(-\frac{2}{3}\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-\frac{11}{3}\right)±\sqrt{\frac{121}{9}+16\left(-\frac{2}{3}\right)}}{2\left(-4\right)}$$

Multiply 16 times -\frac{2}{3}.

$$x=\frac{-\left(-\frac{11}{3}\right)±\sqrt{\frac{121}{9}-\frac{32}{3}}}{2\left(-4\right)}$$

Add \frac{121}{9} to -\frac{32}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{11}{3}\right)±\sqrt{\frac{25}{9}}}{2\left(-4\right)}$$

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

$$x=\frac{-\left(-\frac{11}{3}\right)±\frac{5}{3}}{2\left(-4\right)}$$

The opposite of -\frac{11}{3} is \frac{11}{3}.

$$x=\frac{\frac{11}{3}±\frac{5}{3}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{\frac{11}{3}±\frac{5}{3}}{-8}$$

Now solve the equation x=\frac{\frac{11}{3}±\frac{5}{3}}{-8} when ± is plus. Add \frac{11}{3} to \frac{5}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{16}{3}}{-8}$$

Divide \frac{16}{3} by -8.

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

Now solve the equation x=\frac{\frac{11}{3}±\frac{5}{3}}{-8} when ± is minus. Subtract \frac{5}{3} from \frac{11}{3} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

Reduce the fraction \frac{2}{-8} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

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

Substitute -\frac{2}{3} for x in the equation \sqrt{\frac{x+1}{3}}+5x=3x-1.

$$\sqrt{\frac{-\frac{2}{3}+1}{3}}+5\left(-\frac{2}{3}\right)=3\left(-\frac{2}{3}\right)-1$$

Simplify. The value x=-\frac{2}{3} satisfies the equation.

$$-3=-3$$

Substitute -\frac{1}{4} for x in the equation \sqrt{\frac{x+1}{3}}+5x=3x-1.

$$\sqrt{\frac{-\frac{1}{4}+1}{3}}+5\left(-\frac{1}{4}\right)=3\left(-\frac{1}{4}\right)-1$$

Simplify. The value x=-\frac{1}{4} does not satisfy the equation.

$$-\frac{3}{4}=-\frac{7}{4}$$

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

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