Solve 2p=12+p^2

$$2 p = 12 + p ^ { 2 }$$

Solve for p

$$p=-\sqrt{11}i+1\approx 1-3.31662479i$$

$$p=1+\sqrt{11}i\approx 1+3.31662479i$$

Steps Using the Quadratic Formula

Subtract 12 from both sides.

$$2p-12=p^{2}$$

Subtract p^{2} from both sides.

$$2p-12-p^{2}=0$$

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.

$$-p^{2}+2p-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 2 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\left(-12\right)}}{2\left(-1\right)}$$

Square 2.

$$p=\frac{-2±\sqrt{4-4\left(-1\right)\left(-12\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$p=\frac{-2±\sqrt{4+4\left(-12\right)}}{2\left(-1\right)}$$

Multiply 4 times -12.

$$p=\frac{-2±\sqrt{4-48}}{2\left(-1\right)}$$

Add 4 to -48.

$$p=\frac{-2±\sqrt{-44}}{2\left(-1\right)}$$

Take the square root of -44.

$$p=\frac{-2±2\sqrt{11}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$p=\frac{-2±2\sqrt{11}i}{-2}$$

Now solve the equation p=\frac{-2±2\sqrt{11}i}{-2} when ± is plus. Add -2 to 2i\sqrt{11}.

$$p=\frac{-2+2\sqrt{11}i}{-2}$$

Divide -2+2i\sqrt{11} by -2.

$$p=-\sqrt{11}i+1$$

Now solve the equation p=\frac{-2±2\sqrt{11}i}{-2} when ± is minus. Subtract 2i\sqrt{11} from -2.

$$p=\frac{-2\sqrt{11}i-2}{-2}$$

Divide -2-2i\sqrt{11} by -2.

$$p=1+\sqrt{11}i$$

The equation is now solved.

$$p=-\sqrt{11}i+1 p=1+\sqrt{11}i$$

Steps for Completing the Square

Subtract p^{2} from both sides.

$$2p-p^{2}=12$$

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.

$$-p^{2}+2p=12$$

Divide both sides by -1.

$$\frac{-p^{2}+2p}{-1}=\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$p^{2}+\frac{2}{-1}p=\frac{12}{-1}$$

Divide 2 by -1.

$$p^{2}-2p=\frac{12}{-1}$$

Divide 12 by -1.

$$p^{2}-2p=-12$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-2p+1=-12+1$$

Add -12 to 1.

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

Factor p^{2}-2p+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-1\right)^{2}=-11$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-1\right)^{2}}=\sqrt{-11}$$

Simplify.

$$p-1=\sqrt{11}i p-1=-\sqrt{11}i$$

Add 1 to both sides of the equation.

$$p=1+\sqrt{11}i p=-\sqrt{11}i+1$$