Solve 3w^2+15w+12=w

$$3 w ^ { 2 } + 15 w + 12 = w$$

Solve for w

$$w=\frac{\sqrt{13}-7}{3}\approx -1.131482908$$

$$w=\frac{-\sqrt{13}-7}{3}\approx -3.535183758$$

Steps Using the Quadratic Formula

Subtract w from both sides.

$$3w^{2}+15w+12-w=0$$

Combine 15w and -w to get 14w.

$$3w^{2}+14w+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 14 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-14±\sqrt{14^{2}-4\times 3\times 12}}{2\times 3}$$

Square 14.

$$w=\frac{-14±\sqrt{196-4\times 3\times 12}}{2\times 3}$$

Multiply -4 times 3.

$$w=\frac{-14±\sqrt{196-12\times 12}}{2\times 3}$$

Multiply -12 times 12.

$$w=\frac{-14±\sqrt{196-144}}{2\times 3}$$

Add 196 to -144.

$$w=\frac{-14±\sqrt{52}}{2\times 3}$$

Take the square root of 52.

$$w=\frac{-14±2\sqrt{13}}{2\times 3}$$

Multiply 2 times 3.

$$w=\frac{-14±2\sqrt{13}}{6}$$

Now solve the equation w=\frac{-14±2\sqrt{13}}{6} when ± is plus. Add -14 to 2\sqrt{13}.

$$w=\frac{2\sqrt{13}-14}{6}$$

Divide -14+2\sqrt{13} by 6.

$$w=\frac{\sqrt{13}-7}{3}$$

Now solve the equation w=\frac{-14±2\sqrt{13}}{6} when ± is minus. Subtract 2\sqrt{13} from -14.

$$w=\frac{-2\sqrt{13}-14}{6}$$

Divide -14-2\sqrt{13} by 6.

$$w=\frac{-\sqrt{13}-7}{3}$$

The equation is now solved.

$$w=\frac{\sqrt{13}-7}{3} w=\frac{-\sqrt{13}-7}{3}$$

Steps for Completing the Square

Subtract w from both sides.

$$3w^{2}+15w+12-w=0$$

Combine 15w and -w to get 14w.

$$3w^{2}+14w+12=0$$

Subtract 12 from both sides. Anything subtracted from zero gives its negation.

$$3w^{2}+14w=-12$$

Divide both sides by 3.

$$\frac{3w^{2}+14w}{3}=-\frac{12}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$w^{2}+\frac{14}{3}w=-\frac{12}{3}$$

Divide -12 by 3.

$$w^{2}+\frac{14}{3}w=-4$$

Divide \frac{14}{3}, the coefficient of the x term, by 2 to get \frac{7}{3}. Then add the square of \frac{7}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}+\frac{14}{3}w+\left(\frac{7}{3}\right)^{2}=-4+\left(\frac{7}{3}\right)^{2}$$

Square \frac{7}{3} by squaring both the numerator and the denominator of the fraction.

$$w^{2}+\frac{14}{3}w+\frac{49}{9}=-4+\frac{49}{9}$$

Add -4 to \frac{49}{9}.

$$w^{2}+\frac{14}{3}w+\frac{49}{9}=\frac{13}{9}$$

Factor w^{2}+\frac{14}{3}w+\frac{49}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w+\frac{7}{3}\right)^{2}=\frac{13}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+\frac{7}{3}\right)^{2}}=\sqrt{\frac{13}{9}}$$

Simplify.

$$w+\frac{7}{3}=\frac{\sqrt{13}}{3} w+\frac{7}{3}=-\frac{\sqrt{13}}{3}$$

Subtract \frac{7}{3} from both sides of the equation.

$$w=\frac{\sqrt{13}-7}{3} w=\frac{-\sqrt{13}-7}{3}$$