Solve sqrt{3x+4}=3x+1/2

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

Solve for x

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{3x+4} to the power of 2 and get 3x+4.

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

To raise \frac{3x+1}{2} to a power, raise both numerator and denominator to the power and then divide.

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

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

$$3x+4=\frac{9x^{2}+6x+1}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$3x+4=\frac{9x^{2}+6x+1}{4}$$

Divide each term of 9x^{2}+6x+1 by 4 to get \frac{9}{4}x^{2}+\frac{3}{2}x+\frac{1}{4}.

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

Subtract \frac{9}{4}x^{2} from both sides.

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

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

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

Combine 3x and -\frac{3}{2}x to get \frac{3}{2}x.

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

Subtract \frac{1}{4} from both sides.

$$\frac{3}{2}x+4-\frac{9}{4}x^{2}-\frac{1}{4}=0$$

Subtract \frac{1}{4} from 4 to get \frac{15}{4}.

$$\frac{3}{2}x+\frac{15}{4}-\frac{9}{4}x^{2}=0$$

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.

$$-\frac{9}{4}x^{2}+\frac{3}{2}x+\frac{15}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{9}{4} for a, \frac{3}{2} for b, and \frac{15}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{3}{2}±\sqrt{\left(\frac{3}{2}\right)^{2}-4\left(-\frac{9}{4}\right)\times \frac{15}{4}}}{2\left(-\frac{9}{4}\right)}$$

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

$$x=\frac{-\frac{3}{2}±\sqrt{\frac{9}{4}-4\left(-\frac{9}{4}\right)\times \frac{15}{4}}}{2\left(-\frac{9}{4}\right)}$$

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

$$x=\frac{-\frac{3}{2}±\sqrt{\frac{9}{4}+9\times \frac{15}{4}}}{2\left(-\frac{9}{4}\right)}$$

Multiply 9 times \frac{15}{4}.

$$x=\frac{-\frac{3}{2}±\sqrt{\frac{9+135}{4}}}{2\left(-\frac{9}{4}\right)}$$

Add \frac{9}{4} to \frac{135}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{3}{2}±\sqrt{36}}{2\left(-\frac{9}{4}\right)}$$

Take the square root of 36.

$$x=\frac{-\frac{3}{2}±6}{2\left(-\frac{9}{4}\right)}$$

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

$$x=\frac{-\frac{3}{2}±6}{-\frac{9}{2}}$$

Now solve the equation x=\frac{-\frac{3}{2}±6}{-\frac{9}{2}} when ± is plus. Add -\frac{3}{2} to 6.

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

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

$$x=-1$$

Now solve the equation x=\frac{-\frac{3}{2}±6}{-\frac{9}{2}} when ± is minus. Subtract 6 from -\frac{3}{2}.

$$x=-\frac{\frac{15}{2}}{-\frac{9}{2}}$$

Divide -\frac{15}{2} by -\frac{9}{2} by multiplying -\frac{15}{2} by the reciprocal of -\frac{9}{2}.

$$x=\frac{5}{3}$$

The equation is now solved.

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

Substitute -1 for x in the equation \sqrt{3x+4}=\frac{3x+1}{2}.

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

Simplify. The value x=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Substitute \frac{5}{3} for x in the equation \sqrt{3x+4}=\frac{3x+1}{2}.

$$\sqrt{3\times \frac{5}{3}+4}=\frac{3\times \frac{5}{3}+1}{2}$$

Simplify. The value x=\frac{5}{3} satisfies the equation.

$$3=3$$

Equation \sqrt{3x+4}=\frac{3x+1}{2} has a unique solution.

$$x=\frac{5}{3}$$