Solve 3p^2=-6p+7

$$3 p ^ { 2 } = - 6 p + 7$$

Solve for p

$$p=\frac{\sqrt{30}}{3}-1\approx 0.825741858$$

$$p=-\frac{\sqrt{30}}{3}-1\approx -2.825741858$$

Steps Using the Quadratic Formula

Add 6p to both sides.

$$3p^{2}+6p=7$$

Subtract 7 from both sides.

$$3p^{2}+6p-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 6 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-6±\sqrt{6^{2}-4\times 3\left(-7\right)}}{2\times 3}$$

Square 6.

$$p=\frac{-6±\sqrt{36-4\times 3\left(-7\right)}}{2\times 3}$$

Multiply -4 times 3.

$$p=\frac{-6±\sqrt{36-12\left(-7\right)}}{2\times 3}$$

Multiply -12 times -7.

$$p=\frac{-6±\sqrt{36+84}}{2\times 3}$$

Add 36 to 84.

$$p=\frac{-6±\sqrt{120}}{2\times 3}$$

Take the square root of 120.

$$p=\frac{-6±2\sqrt{30}}{2\times 3}$$

Multiply 2 times 3.

$$p=\frac{-6±2\sqrt{30}}{6}$$

Now solve the equation p=\frac{-6±2\sqrt{30}}{6} when ± is plus. Add -6 to 2\sqrt{30}.

$$p=\frac{2\sqrt{30}-6}{6}$$

Divide -6+2\sqrt{30} by 6.

$$p=\frac{\sqrt{30}}{3}-1$$

Now solve the equation p=\frac{-6±2\sqrt{30}}{6} when ± is minus. Subtract 2\sqrt{30} from -6.

$$p=\frac{-2\sqrt{30}-6}{6}$$

Divide -6-2\sqrt{30} by 6.

$$p=-\frac{\sqrt{30}}{3}-1$$

The equation is now solved.

$$p=\frac{\sqrt{30}}{3}-1 p=-\frac{\sqrt{30}}{3}-1$$

Steps for Completing the Square

Add 6p to both sides.

$$3p^{2}+6p=7$$

Divide both sides by 3.

$$\frac{3p^{2}+6p}{3}=\frac{7}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$p^{2}+\frac{6}{3}p=\frac{7}{3}$$

Divide 6 by 3.

$$p^{2}+2p=\frac{7}{3}$$

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.

$$p^{2}+2p+1^{2}=\frac{7}{3}+1^{2}$$

Square 1.

$$p^{2}+2p+1=\frac{7}{3}+1$$

Add \frac{7}{3} to 1.

$$p^{2}+2p+1=\frac{10}{3}$$

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}.

$$\left(p+1\right)^{2}=\frac{10}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p+1\right)^{2}}=\sqrt{\frac{10}{3}}$$

Simplify.

$$p+1=\frac{\sqrt{30}}{3} p+1=-\frac{\sqrt{30}}{3}$$

Subtract 1 from both sides of the equation.

$$p=\frac{\sqrt{30}}{3}-1 p=-\frac{\sqrt{30}}{3}-1$$