Solve 3w^2-12w=-12quad14

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

Solve for w

$$w=2+2\sqrt{13}i\approx 2+7.211102551i$$

$$w=-2\sqrt{13}i+2\approx 2-7.211102551i$$

Steps Using the Quadratic Formula

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.

$$3w^{2}-12w=-168$$

Add 168 to both sides of the equation.

$$3w^{2}-12w-\left(-168\right)=-168-\left(-168\right)$$

Subtracting -168 from itself leaves 0.

$$3w^{2}-12w-\left(-168\right)=0$$

Subtract -168 from 0.

$$3w^{2}-12w+168=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -12 for b, and 168 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 3\times 168}}{2\times 3}$$

Square -12.

$$w=\frac{-\left(-12\right)±\sqrt{144-4\times 3\times 168}}{2\times 3}$$

Multiply -4 times 3.

$$w=\frac{-\left(-12\right)±\sqrt{144-12\times 168}}{2\times 3}$$

Multiply -12 times 168.

$$w=\frac{-\left(-12\right)±\sqrt{144-2016}}{2\times 3}$$

Add 144 to -2016.

$$w=\frac{-\left(-12\right)±\sqrt{-1872}}{2\times 3}$$

Take the square root of -1872.

$$w=\frac{-\left(-12\right)±12\sqrt{13}i}{2\times 3}$$

The opposite of -12 is 12.

$$w=\frac{12±12\sqrt{13}i}{2\times 3}$$

Multiply 2 times 3.

$$w=\frac{12±12\sqrt{13}i}{6}$$

Now solve the equation w=\frac{12±12\sqrt{13}i}{6} when ± is plus. Add 12 to 12i\sqrt{13}.

$$w=\frac{12+12\sqrt{13}i}{6}$$

Divide 12+12i\sqrt{13} by 6.

$$w=2+2\sqrt{13}i$$

Now solve the equation w=\frac{12±12\sqrt{13}i}{6} when ± is minus. Subtract 12i\sqrt{13} from 12.

$$w=\frac{-12\sqrt{13}i+12}{6}$$

Divide 12-12i\sqrt{13} by 6.

$$w=-2\sqrt{13}i+2$$

The equation is now solved.

$$w=2+2\sqrt{13}i w=-2\sqrt{13}i+2$$

Steps for Completing the Square

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.

$$3w^{2}-12w=-168$$

Divide both sides by 3.

$$\frac{3w^{2}-12w}{3}=-\frac{168}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$w^{2}+\left(-\frac{12}{3}\right)w=-\frac{168}{3}$$

Divide -12 by 3.

$$w^{2}-4w=-\frac{168}{3}$$

Divide -168 by 3.

$$w^{2}-4w=-56$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}-4w+\left(-2\right)^{2}=-56+\left(-2\right)^{2}$$

Square -2.

$$w^{2}-4w+4=-56+4$$

Add -56 to 4.

$$w^{2}-4w+4=-52$$

Factor w^{2}-4w+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w-2\right)^{2}=-52$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-2\right)^{2}}=\sqrt{-52}$$

Simplify.

$$w-2=2\sqrt{13}i w-2=-2\sqrt{13}i$$

Add 2 to both sides of the equation.

$$w=2+2\sqrt{13}i w=-2\sqrt{13}i+2$$