Solve p^2=-6p-1

$$p ^ { 2 } = - 6 p - 1$$

Solve for p

$$p=2\sqrt{2}-3\approx -0.171572875$$

$$p=-2\sqrt{2}-3\approx -5.828427125$$

Steps Using the Quadratic Formula

Add 6p to both sides.

$$p^{2}+6p=-1$$

Add 1 to both sides.

$$p^{2}+6p+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-6±\sqrt{6^{2}-4}}{2}$$

Square 6.

$$p=\frac{-6±\sqrt{36-4}}{2}$$

Add 36 to -4.

$$p=\frac{-6±\sqrt{32}}{2}$$

Take the square root of 32.

$$p=\frac{-6±4\sqrt{2}}{2}$$

Now solve the equation p=\frac{-6±4\sqrt{2}}{2} when ± is plus. Add -6 to 4\sqrt{2}.

$$p=\frac{4\sqrt{2}-6}{2}$$

Divide -6+4\sqrt{2} by 2.

$$p=2\sqrt{2}-3$$

Now solve the equation p=\frac{-6±4\sqrt{2}}{2} when ± is minus. Subtract 4\sqrt{2} from -6.

$$p=\frac{-4\sqrt{2}-6}{2}$$

Divide -6-4\sqrt{2} by 2.

$$p=-2\sqrt{2}-3$$

The equation is now solved.

$$p=2\sqrt{2}-3 p=-2\sqrt{2}-3$$

Steps for Completing the Square

Add 6p to both sides.

$$p^{2}+6p=-1$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}+6p+3^{2}=-1+3^{2}$$

Square 3.

$$p^{2}+6p+9=-1+9$$

Add -1 to 9.

$$p^{2}+6p+9=8$$

Factor p^{2}+6p+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p+3\right)^{2}=8$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+3\right)^{2}}=\sqrt{8}$$

Simplify.

$$p+3=2\sqrt{2} p+3=-2\sqrt{2}$$

Subtract 3 from both sides of the equation.

$$p=2\sqrt{2}-3 p=-2\sqrt{2}-3$$