Solve -34=p(p-12)+2

$$- 34 = p ( p - 12 ) + 2$$

Solve for p

$$p=6$$

Steps Using the Quadratic Formula

Use the distributive property to multiply p by p-12.

$$-34=p^{2}-12p+2$$

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

$$p^{2}-12p+2=-34$$

Add 34 to both sides.

$$p^{2}-12p+2+34=0$$

Add 2 and 34 to get 36.

$$p^{2}-12p+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 36}}{2}$$

Square -12.

$$p=\frac{-\left(-12\right)±\sqrt{144-4\times 36}}{2}$$

Multiply -4 times 36.

$$p=\frac{-\left(-12\right)±\sqrt{144-144}}{2}$$

Add 144 to -144.

$$p=\frac{-\left(-12\right)±\sqrt{0}}{2}$$

Take the square root of 0.

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

The opposite of -12 is 12.

$$p=\frac{12}{2}$$

Divide 12 by 2.

$$p=6$$

Steps for Completing the Square

Use the distributive property to multiply p by p-12.

$$-34=p^{2}-12p+2$$

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

$$p^{2}-12p+2=-34$$

Subtract 2 from both sides.

$$p^{2}-12p=-34-2$$

Subtract 2 from -34 to get -36.

$$p^{2}-12p=-36$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-12p+\left(-6\right)^{2}=-36+\left(-6\right)^{2}$$

Square -6.

$$p^{2}-12p+36=-36+36$$

Add -36 to 36.

$$p^{2}-12p+36=0$$

Factor p^{2}-12p+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-6\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-6\right)^{2}}=\sqrt{0}$$

Simplify.

$$p-6=0 p-6=0$$

Add 6 to both sides of the equation.

$$p=6 p=6$$

The equation is now solved. Solutions are the same.

$$p=6$$