Variable $x$ cannot be equal to $-\frac{1}{2}$ since division by zero is not defined. Multiply both sides of the equation by $2x+1$.
$$2x+1+3x\left(2x+1\right)=-3\left(2x+1\right)$$
Use the distributive property to multiply $3x$ by $2x+1$.
$$2x+1+6x^{2}+3x=-3\left(2x+1\right)$$
Combine $2x$ and $3x$ to get $5x$.
$$5x+1+6x^{2}=-3\left(2x+1\right)$$
Use the distributive property to multiply $-3$ by $2x+1$.
$$5x+1+6x^{2}=-6x-3$$
Add $6x$ to both sides.
$$5x+1+6x^{2}+6x=-3$$
Combine $5x$ and $6x$ to get $11x$.
$$11x+1+6x^{2}=-3$$
Add $3$ to both sides.
$$11x+1+6x^{2}+3=0$$
Add $1$ and $3$ to get $4$.
$$11x+4+6x^{2}=0$$
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.
$$6x^{2}+11x+4=0$$
This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $6$ for $a$, $11$ for $b$, and $4$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.
Now solve the equation $x=\frac{-11±5}{12}$ when $±$ is plus. Add $-11$ to $5$.
$$x=-\frac{6}{12}$$
Reduce the fraction $\frac{-6}{12}$ to lowest terms by extracting and canceling out $6$.
$$x=-\frac{1}{2}$$
Now solve the equation $x=\frac{-11±5}{12}$ when $±$ is minus. Subtract $5$ from $-11$.
$$x=-\frac{16}{12}$$
Reduce the fraction $\frac{-16}{12}$ to lowest terms by extracting and canceling out $4$.
$$x=-\frac{4}{3}$$
The equation is now solved.
$$x=-\frac{1}{2}$$ $$x=-\frac{4}{3}$$
Variable $x$ cannot be equal to $-\frac{1}{2}$.
$$x=-\frac{4}{3}$$
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 $2x+1$.
$$2x+1+3x\left(2x+1\right)=-3\left(2x+1\right)$$
Use the distributive property to multiply $3x$ by $2x+1$.
$$2x+1+6x^{2}+3x=-3\left(2x+1\right)$$
Combine $2x$ and $3x$ to get $5x$.
$$5x+1+6x^{2}=-3\left(2x+1\right)$$
Use the distributive property to multiply $-3$ by $2x+1$.
$$5x+1+6x^{2}=-6x-3$$
Add $6x$ to both sides.
$$5x+1+6x^{2}+6x=-3$$
Combine $5x$ and $6x$ to get $11x$.
$$11x+1+6x^{2}=-3$$
Subtract $1$ from both sides.
$$11x+6x^{2}=-3-1$$
Subtract $1$ from $-3$ to get $-4$.
$$11x+6x^{2}=-4$$
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form $x^{2}+bx=c$.
$$6x^{2}+11x=-4$$
Divide both sides by $6$.
$$\frac{6x^{2}+11x}{6}=-\frac{4}{6}$$
Dividing by $6$ undoes the multiplication by $6$.
$$x^{2}+\frac{11}{6}x=-\frac{4}{6}$$
Reduce the fraction $\frac{-4}{6}$ to lowest terms by extracting and canceling out $2$.
$$x^{2}+\frac{11}{6}x=-\frac{2}{3}$$
Divide $\frac{11}{6}$, the coefficient of the $x$ term, by $2$ to get $\frac{11}{12}$. Then add the square of $\frac{11}{12}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.
Add $-\frac{2}{3}$ to $\frac{121}{144}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
Factor $x^{2}+\frac{11}{6}x+\frac{121}{144}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.
