Solve 3x+1/2+1/3(2x+1)=0

$$\frac{ 3x+1 }{ 2 } + \frac{ 1 }{ 3(2x+1) } =0$$

Solve for x (complex solution)

$$x=\frac{-5+\sqrt{15}i}{12}\approx -0.416666667+0.322748612i$$

$$x=\frac{-\sqrt{15}i-5}{12}\approx -0.416666667-0.322748612i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 6\left(2x+1\right), the least common multiple of 2,3\left(2x+1\right).

$$\left(6x+3\right)\left(3x+1\right)+2=0$$

Use the distributive property to multiply 6x+3 by 3x+1 and combine like terms.

$$18x^{2}+15x+3+2=0$$

Add 3 and 2 to get 5.

$$18x^{2}+15x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 18 for a, 15 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 18\times 5}}{2\times 18}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 18\times 5}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-15±\sqrt{225-72\times 5}}{2\times 18}$$

Multiply -72 times 5.

$$x=\frac{-15±\sqrt{225-360}}{2\times 18}$$

Add 225 to -360.

$$x=\frac{-15±\sqrt{-135}}{2\times 18}$$

Take the square root of -135.

$$x=\frac{-15±3\sqrt{15}i}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{-15±3\sqrt{15}i}{36}$$

Now solve the equation x=\frac{-15±3\sqrt{15}i}{36} when ± is plus. Add -15 to 3i\sqrt{15}.

$$x=\frac{-15+3\sqrt{15}i}{36}$$

Divide -15+3i\sqrt{15} by 36.

$$x=\frac{-5+\sqrt{15}i}{12}$$

Now solve the equation x=\frac{-15±3\sqrt{15}i}{36} when ± is minus. Subtract 3i\sqrt{15} from -15.

$$x=\frac{-3\sqrt{15}i-15}{36}$$

Divide -15-3i\sqrt{15} by 36.

$$x=\frac{-\sqrt{15}i-5}{12}$$

The equation is now solved.

$$x=\frac{-5+\sqrt{15}i}{12} x=\frac{-\sqrt{15}i-5}{12}$$

Steps for Completing the Square

Variable x cannot be equal to -\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 6\left(2x+1\right), the least common multiple of 2,3\left(2x+1\right).

$$\left(6x+3\right)\left(3x+1\right)+2=0$$

Use the distributive property to multiply 6x+3 by 3x+1 and combine like terms.

$$18x^{2}+15x+3+2=0$$

Add 3 and 2 to get 5.

$$18x^{2}+15x+5=0$$

Subtract 5 from both sides. Anything subtracted from zero gives its negation.

$$18x^{2}+15x=-5$$

Divide both sides by 18.

$$\frac{18x^{2}+15x}{18}=-\frac{5}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$x^{2}+\frac{15}{18}x=-\frac{5}{18}$$

Reduce the fraction \frac{15}{18} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{5}{6}x=-\frac{5}{18}$$

Divide \frac{5}{6}, the coefficient of the x term, by 2 to get \frac{5}{12}. Then add the square of \frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{6}x+\left(\frac{5}{12}\right)^{2}=-\frac{5}{18}+\left(\frac{5}{12}\right)^{2}$$

Square \frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{6}x+\frac{25}{144}=-\frac{5}{18}+\frac{25}{144}$$

Add -\frac{5}{18} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{6}x+\frac{25}{144}=-\frac{5}{48}$$

Factor x^{2}+\frac{5}{6}x+\frac{25}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{12}\right)^{2}=-\frac{5}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{12}\right)^{2}}=\sqrt{-\frac{5}{48}}$$

Simplify.

$$x+\frac{5}{12}=\frac{\sqrt{15}i}{12} x+\frac{5}{12}=-\frac{\sqrt{15}i}{12}$$

Subtract \frac{5}{12} from both sides of the equation.

$$x=\frac{-5+\sqrt{15}i}{12} x=\frac{-\sqrt{15}i-5}{12}$$