Solve 12x(x+1)=x-2

$$12 x ( x + 1 ) = x - 2$$

Solve for x

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

$$x=-\frac{1}{4}=-0.25$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 12x by x+1.

$$12x^{2}+12x=x-2$$

Subtract x from both sides.

$$12x^{2}+12x-x=-2$$

Combine 12x and -x to get 11x.

$$12x^{2}+11x=-2$$

Add 2 to both sides.

$$12x^{2}+11x+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 11 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 12\times 2}}{2\times 12}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 12\times 2}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-11±\sqrt{121-48\times 2}}{2\times 12}$$

Multiply -48 times 2.

$$x=\frac{-11±\sqrt{121-96}}{2\times 12}$$

Add 121 to -96.

$$x=\frac{-11±\sqrt{25}}{2\times 12}$$

Take the square root of 25.

$$x=\frac{-11±5}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-11±5}{24}$$

Now solve the equation x=\frac{-11±5}{24} when ± is plus. Add -11 to 5.

$$x=-\frac{6}{24}$$

Reduce the fraction \frac{-6}{24} to lowest terms by extracting and canceling out 6.

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

Now solve the equation x=\frac{-11±5}{24} when ± is minus. Subtract 5 from -11.

$$x=-\frac{16}{24}$$

Reduce the fraction \frac{-16}{24} to lowest terms by extracting and canceling out 8.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 12x by x+1.

$$12x^{2}+12x=x-2$$

Subtract x from both sides.

$$12x^{2}+12x-x=-2$$

Combine 12x and -x to get 11x.

$$12x^{2}+11x=-2$$

Divide both sides by 12.

$$\frac{12x^{2}+11x}{12}=-\frac{2}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{11}{12}x=-\frac{2}{12}$$

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

$$x^{2}+\frac{11}{12}x=-\frac{1}{6}$$

Divide \frac{11}{12}, the coefficient of the x term, by 2 to get \frac{11}{24}. Then add the square of \frac{11}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{12}x+\left(\frac{11}{24}\right)^{2}=-\frac{1}{6}+\left(\frac{11}{24}\right)^{2}$$

Square \frac{11}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{12}x+\frac{121}{576}=-\frac{1}{6}+\frac{121}{576}$$

Add -\frac{1}{6} to \frac{121}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{11}{12}x+\frac{121}{576}=\frac{25}{576}$$

Factor x^{2}+\frac{11}{12}x+\frac{121}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{11}{24}\right)^{2}=\frac{25}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{24}\right)^{2}}=\sqrt{\frac{25}{576}}$$

Simplify.

$$x+\frac{11}{24}=\frac{5}{24} x+\frac{11}{24}=-\frac{5}{24}$$

Subtract \frac{11}{24} from both sides of the equation.

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