Solve -3p^2-300p+112700=0

$$-3 { p }^{ 2 } -300p+112700 = 0$$

Solve for p

$$p=\frac{10\sqrt{3606}}{3}-50\approx 150.16659728$$

$$p=-\frac{10\sqrt{3606}}{3}-50\approx -250.16659728$$

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.

$$-3p^{2}-300p+112700=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -300 for b, and 112700 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-300\right)±\sqrt{\left(-300\right)^{2}-4\left(-3\right)\times 112700}}{2\left(-3\right)}$$

Square -300.

$$p=\frac{-\left(-300\right)±\sqrt{90000-4\left(-3\right)\times 112700}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$p=\frac{-\left(-300\right)±\sqrt{90000+12\times 112700}}{2\left(-3\right)}$$

Multiply 12 times 112700.

$$p=\frac{-\left(-300\right)±\sqrt{90000+1352400}}{2\left(-3\right)}$$

Add 90000 to 1352400.

$$p=\frac{-\left(-300\right)±\sqrt{1442400}}{2\left(-3\right)}$$

Take the square root of 1442400.

$$p=\frac{-\left(-300\right)±20\sqrt{3606}}{2\left(-3\right)}$$

The opposite of -300 is 300.

$$p=\frac{300±20\sqrt{3606}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$p=\frac{300±20\sqrt{3606}}{-6}$$

Now solve the equation p=\frac{300±20\sqrt{3606}}{-6} when ± is plus. Add 300 to 20\sqrt{3606}.

$$p=\frac{20\sqrt{3606}+300}{-6}$$

Divide 300+20\sqrt{3606} by -6.

$$p=-\frac{10\sqrt{3606}}{3}-50$$

Now solve the equation p=\frac{300±20\sqrt{3606}}{-6} when ± is minus. Subtract 20\sqrt{3606} from 300.

$$p=\frac{300-20\sqrt{3606}}{-6}$$

Divide 300-20\sqrt{3606} by -6.

$$p=\frac{10\sqrt{3606}}{3}-50$$

The equation is now solved.

$$p=-\frac{10\sqrt{3606}}{3}-50 p=\frac{10\sqrt{3606}}{3}-50$$

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.

$$-3p^{2}-300p+112700=0$$

Subtract 112700 from both sides of the equation.

$$-3p^{2}-300p+112700-112700=-112700$$

Subtracting 112700 from itself leaves 0.

$$-3p^{2}-300p=-112700$$

Divide both sides by -3.

$$\frac{-3p^{2}-300p}{-3}=-\frac{112700}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$p^{2}+\left(-\frac{300}{-3}\right)p=-\frac{112700}{-3}$$

Divide -300 by -3.

$$p^{2}+100p=-\frac{112700}{-3}$$

Divide -112700 by -3.

$$p^{2}+100p=\frac{112700}{3}$$

Divide 100, the coefficient of the x term, by 2 to get 50. Then add the square of 50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+100p+50^{2}=\frac{112700}{3}+50^{2}$$

Square 50.

$$p^{2}+100p+2500=\frac{112700}{3}+2500$$

Add \frac{112700}{3} to 2500.

$$p^{2}+100p+2500=\frac{120200}{3}$$

Factor p^{2}+100p+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+50\right)^{2}=\frac{120200}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+50\right)^{2}}=\sqrt{\frac{120200}{3}}$$

Simplify.

$$p+50=\frac{10\sqrt{3606}}{3} p+50=-\frac{10\sqrt{3606}}{3}$$

Subtract 50 from both sides of the equation.

$$p=\frac{10\sqrt{3606}}{3}-50 p=-\frac{10\sqrt{3606}}{3}-50$$