Solve 2100=-3p^2+132p

$$2100 = - 3 p ^ { 2 } + 132 p$$

Solve for p

$$p=22+6\sqrt{6}i\approx 22+14.696938457i$$

$$p=-6\sqrt{6}i+22\approx 22-14.696938457i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-3p^{2}+132p=2100$$

Subtract 2100 from both sides.

$$-3p^{2}+132p-2100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 132 for b, and -2100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-132±\sqrt{132^{2}-4\left(-3\right)\left(-2100\right)}}{2\left(-3\right)}$$

Square 132.

$$p=\frac{-132±\sqrt{17424-4\left(-3\right)\left(-2100\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$p=\frac{-132±\sqrt{17424+12\left(-2100\right)}}{2\left(-3\right)}$$

Multiply 12 times -2100.

$$p=\frac{-132±\sqrt{17424-25200}}{2\left(-3\right)}$$

Add 17424 to -25200.

$$p=\frac{-132±\sqrt{-7776}}{2\left(-3\right)}$$

Take the square root of -7776.

$$p=\frac{-132±36\sqrt{6}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$p=\frac{-132±36\sqrt{6}i}{-6}$$

Now solve the equation p=\frac{-132±36\sqrt{6}i}{-6} when ± is plus. Add -132 to 36i\sqrt{6}.

$$p=\frac{-132+36\sqrt{6}i}{-6}$$

Divide -132+36i\sqrt{6} by -6.

$$p=-6\sqrt{6}i+22$$

Now solve the equation p=\frac{-132±36\sqrt{6}i}{-6} when ± is minus. Subtract 36i\sqrt{6} from -132.

$$p=\frac{-36\sqrt{6}i-132}{-6}$$

Divide -132-36i\sqrt{6} by -6.

$$p=22+6\sqrt{6}i$$

The equation is now solved.

$$p=-6\sqrt{6}i+22 p=22+6\sqrt{6}i$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-3p^{2}+132p=2100$$

Divide both sides by -3.

$$\frac{-3p^{2}+132p}{-3}=\frac{2100}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$p^{2}+\frac{132}{-3}p=\frac{2100}{-3}$$

Divide 132 by -3.

$$p^{2}-44p=\frac{2100}{-3}$$

Divide 2100 by -3.

$$p^{2}-44p=-700$$

Divide -44, the coefficient of the x term, by 2 to get -22. Then add the square of -22 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-44p+\left(-22\right)^{2}=-700+\left(-22\right)^{2}$$

Square -22.

$$p^{2}-44p+484=-700+484$$

Add -700 to 484.

$$p^{2}-44p+484=-216$$

Factor p^{2}-44p+484. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-22\right)^{2}=-216$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-22\right)^{2}}=\sqrt{-216}$$

Simplify.

$$p-22=6\sqrt{6}i p-22=-6\sqrt{6}i$$

Add 22 to both sides of the equation.

$$p=22+6\sqrt{6}i p=-6\sqrt{6}i+22$$