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

$$(3x+2)(2x+1)=1$$

Solve for x

$$x=-\frac{1}{6}\approx -0.166666667$$

$$x=-1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x+2 by 2x+1 and combine like terms.

$$6x^{2}+7x+2=1$$

Subtract 1 from both sides.

$$6x^{2}+7x+2-1=0$$

Subtract 1 from 2 to get 1.

$$6x^{2}+7x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 7 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\times 6}}{2\times 6}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 6}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-7±\sqrt{49-24}}{2\times 6}$$

Add 49 to -24.

$$x=\frac{-7±\sqrt{25}}{2\times 6}$$

Take the square root of 25.

$$x=\frac{-7±5}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-7±5}{12}$$

Now solve the equation x=\frac{-7±5}{12} when ± is plus. Add -7 to 5.

$$x=-\frac{2}{12}$$

Reduce the fraction \frac{-2}{12} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{6}$$

Now solve the equation x=\frac{-7±5}{12} when ± is minus. Subtract 5 from -7.

$$x=-\frac{12}{12}$$

Divide -12 by 12.

$$x=-1$$

The equation is now solved.

$$x=-\frac{1}{6} x=-1$$

Steps for Completing the Square

Use the distributive property to multiply 3x+2 by 2x+1 and combine like terms.

$$6x^{2}+7x+2=1$$

Subtract 2 from both sides.

$$6x^{2}+7x=1-2$$

Subtract 2 from 1 to get -1.

$$6x^{2}+7x=-1$$

Divide both sides by 6.

$$\frac{6x^{2}+7x}{6}=-\frac{1}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{7}{6}x=-\frac{1}{6}$$

Divide \frac{7}{6}, the coefficient of the x term, by 2 to get \frac{7}{12}. Then add the square of \frac{7}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{6}x+\left(\frac{7}{12}\right)^{2}=-\frac{1}{6}+\left(\frac{7}{12}\right)^{2}$$

Square \frac{7}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=-\frac{1}{6}+\frac{49}{144}$$

Add -\frac{1}{6} to \frac{49}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{6}x+\frac{49}{144}=\frac{25}{144}$$

Factor x^{2}+\frac{7}{6}x+\frac{49}{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}.

$$\left(x+\frac{7}{12}\right)^{2}=\frac{25}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{12}\right)^{2}}=\sqrt{\frac{25}{144}}$$

Simplify.

$$x+\frac{7}{12}=\frac{5}{12} x+\frac{7}{12}=-\frac{5}{12}$$

Subtract \frac{7}{12} from both sides of the equation.

$$x=-\frac{1}{6} x=-1$$