Solve -3p^2-66p=3

$$- 3 p ^ { 2 } - 66 p = 3$$

Solve for p

$$p=2\sqrt{30}-11\approx -0.04554885$$

$$p=-2\sqrt{30}-11\approx -21.95445115$$

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.

$$-3p^{2}-66p=3$$

Subtract 3 from both sides of the equation.

$$-3p^{2}-66p-3=3-3$$

Subtracting 3 from itself leaves 0.

$$-3p^{2}-66p-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -66 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-66\right)±\sqrt{\left(-66\right)^{2}-4\left(-3\right)\left(-3\right)}}{2\left(-3\right)}$$

Square -66.

$$p=\frac{-\left(-66\right)±\sqrt{4356-4\left(-3\right)\left(-3\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$p=\frac{-\left(-66\right)±\sqrt{4356+12\left(-3\right)}}{2\left(-3\right)}$$

Multiply 12 times -3.

$$p=\frac{-\left(-66\right)±\sqrt{4356-36}}{2\left(-3\right)}$$

Add 4356 to -36.

$$p=\frac{-\left(-66\right)±\sqrt{4320}}{2\left(-3\right)}$$

Take the square root of 4320.

$$p=\frac{-\left(-66\right)±12\sqrt{30}}{2\left(-3\right)}$$

The opposite of -66 is 66.

$$p=\frac{66±12\sqrt{30}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$p=\frac{66±12\sqrt{30}}{-6}$$

Now solve the equation p=\frac{66±12\sqrt{30}}{-6} when ± is plus. Add 66 to 12\sqrt{30}.

$$p=\frac{12\sqrt{30}+66}{-6}$$

Divide 66+12\sqrt{30} by -6.

$$p=-2\sqrt{30}-11$$

Now solve the equation p=\frac{66±12\sqrt{30}}{-6} when ± is minus. Subtract 12\sqrt{30} from 66.

$$p=\frac{66-12\sqrt{30}}{-6}$$

Divide 66-12\sqrt{30} by -6.

$$p=2\sqrt{30}-11$$

The equation is now solved.

$$p=-2\sqrt{30}-11 p=2\sqrt{30}-11$$

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.

$$-3p^{2}-66p=3$$

Divide both sides by -3.

$$\frac{-3p^{2}-66p}{-3}=\frac{3}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$p^{2}+\left(-\frac{66}{-3}\right)p=\frac{3}{-3}$$

Divide -66 by -3.

$$p^{2}+22p=\frac{3}{-3}$$

Divide 3 by -3.

$$p^{2}+22p=-1$$

Divide 22, the coefficient of the x term, by 2 to get 11. Then add the square of 11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+22p+11^{2}=-1+11^{2}$$

Square 11.

$$p^{2}+22p+121=-1+121$$

Add -1 to 121.

$$p^{2}+22p+121=120$$

Factor p^{2}+22p+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+11\right)^{2}=120$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+11\right)^{2}}=\sqrt{120}$$

Simplify.

$$p+11=2\sqrt{30} p+11=-2\sqrt{30}$$

Subtract 11 from both sides of the equation.

$$p=2\sqrt{30}-11 p=-2\sqrt{30}-11$$