Solve 3P=(1-3P^2)+1

$$3 P = ( 1 - 3 P ^ { 2 } ) + 1$$

Solve for P

$$P=\frac{\sqrt{33}}{6}-\frac{1}{2}\approx 0.457427108$$

$$P=-\frac{\sqrt{33}}{6}-\frac{1}{2}\approx -1.457427108$$

Steps Using the Quadratic Formula

Add 1 and 1 to get 2.

$$3P=2-3P^{2}$$

Subtract 2 from both sides.

$$3P-2=-3P^{2}$$

Add 3P^{2} to both sides.

$$3P-2+3P^{2}=0$$

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.

$$3P^{2}+3P-2=0$$

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

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

Square 3.

$$P=\frac{-3±\sqrt{9-4\times 3\left(-2\right)}}{2\times 3}$$

Multiply -4 times 3.

$$P=\frac{-3±\sqrt{9-12\left(-2\right)}}{2\times 3}$$

Multiply -12 times -2.

$$P=\frac{-3±\sqrt{9+24}}{2\times 3}$$

Add 9 to 24.

$$P=\frac{-3±\sqrt{33}}{2\times 3}$$

Multiply 2 times 3.

$$P=\frac{-3±\sqrt{33}}{6}$$

Now solve the equation P=\frac{-3±\sqrt{33}}{6} when ± is plus. Add -3 to \sqrt{33}.

$$P=\frac{\sqrt{33}-3}{6}$$

Divide -3+\sqrt{33} by 6.

$$P=\frac{\sqrt{33}}{6}-\frac{1}{2}$$

Now solve the equation P=\frac{-3±\sqrt{33}}{6} when ± is minus. Subtract \sqrt{33} from -3.

$$P=\frac{-\sqrt{33}-3}{6}$$

Divide -3-\sqrt{33} by 6.

$$P=-\frac{\sqrt{33}}{6}-\frac{1}{2}$$

The equation is now solved.

$$P=\frac{\sqrt{33}}{6}-\frac{1}{2} P=-\frac{\sqrt{33}}{6}-\frac{1}{2}$$

Steps for Completing the Square

Add 1 and 1 to get 2.

$$3P=2-3P^{2}$$

Add 3P^{2} to both sides.

$$3P+3P^{2}=2$$

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.

$$3P^{2}+3P=2$$

Divide both sides by 3.

$$\frac{3P^{2}+3P}{3}=\frac{2}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$P^{2}+\frac{3}{3}P=\frac{2}{3}$$

Divide 3 by 3.

$$P^{2}+P=\frac{2}{3}$$

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

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$P^{2}+P+\frac{1}{4}=\frac{2}{3}+\frac{1}{4}$$

Add \frac{2}{3} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$P^{2}+P+\frac{1}{4}=\frac{11}{12}$$

Factor P^{2}+P+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(P+\frac{1}{2}\right)^{2}=\frac{11}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(P+\frac{1}{2}\right)^{2}}=\sqrt{\frac{11}{12}}$$

Simplify.

$$P+\frac{1}{2}=\frac{\sqrt{33}}{6} P+\frac{1}{2}=-\frac{\sqrt{33}}{6}$$

Subtract \frac{1}{2} from both sides of the equation.

$$P=\frac{\sqrt{33}}{6}-\frac{1}{2} P=-\frac{\sqrt{33}}{6}-\frac{1}{2}$$