Solve 3.6*p*(1-p)=0.96

$$3.6 \times p \times ( 1 - p ) = 0.96$$

Solve for p

$$p=\frac{i\sqrt{15}}{30}+0.5\approx 0.5+0.129099445i$$

$$p=-\frac{i\sqrt{15}}{30}+0.5\approx 0.5-0.129099445i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3.6p by 1-p.

$$3.6p-3.6p^{2}=0.96$$

Subtract 0.96 from both sides.

$$3.6p-3.6p^{2}-0.96=0$$

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.

$$-3.6p^{2}+3.6p-0.96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3.6 for a, 3.6 for b, and -0.96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-3.6±\sqrt{3.6^{2}-4\left(-3.6\right)\left(-0.96\right)}}{2\left(-3.6\right)}$$

Square 3.6 by squaring both the numerator and the denominator of the fraction.

$$p=\frac{-3.6±\sqrt{12.96-4\left(-3.6\right)\left(-0.96\right)}}{2\left(-3.6\right)}$$

Multiply -4 times -3.6.

$$p=\frac{-3.6±\sqrt{12.96+14.4\left(-0.96\right)}}{2\left(-3.6\right)}$$

Multiply 14.4 times -0.96 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$p=\frac{-3.6±\sqrt{12.96-13.824}}{2\left(-3.6\right)}$$

Add 12.96 to -13.824 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p=\frac{-3.6±\sqrt{-0.864}}{2\left(-3.6\right)}$$

Take the square root of -0.864.

$$p=\frac{-3.6±\frac{6\sqrt{15}i}{25}}{2\left(-3.6\right)}$$

Multiply 2 times -3.6.

$$p=\frac{-3.6±\frac{6\sqrt{15}i}{25}}{-7.2}$$

Now solve the equation p=\frac{-3.6±\frac{6\sqrt{15}i}{25}}{-7.2} when ± is plus. Add -3.6 to \frac{6i\sqrt{15}}{25}.

$$p=\frac{\frac{6\sqrt{15}i}{25}-\frac{18}{5}}{-7.2}$$

Divide -\frac{18}{5}+\frac{6i\sqrt{15}}{25} by -7.2 by multiplying -\frac{18}{5}+\frac{6i\sqrt{15}}{25} by the reciprocal of -7.2.

$$p=-\frac{\sqrt{15}i}{30}+\frac{1}{2}$$

Now solve the equation p=\frac{-3.6±\frac{6\sqrt{15}i}{25}}{-7.2} when ± is minus. Subtract \frac{6i\sqrt{15}}{25} from -3.6.

$$p=\frac{-\frac{6\sqrt{15}i}{25}-\frac{18}{5}}{-7.2}$$

Divide -\frac{18}{5}-\frac{6i\sqrt{15}}{25} by -7.2 by multiplying -\frac{18}{5}-\frac{6i\sqrt{15}}{25} by the reciprocal of -7.2.

$$p=\frac{\sqrt{15}i}{30}+\frac{1}{2}$$

The equation is now solved.

$$p=-\frac{\sqrt{15}i}{30}+\frac{1}{2} p=\frac{\sqrt{15}i}{30}+\frac{1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 3.6p by 1-p.

$$3.6p-3.6p^{2}=0.96$$

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.

$$-3.6p^{2}+3.6p=0.96$$

Divide both sides of the equation by -3.6, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-3.6p^{2}+3.6p}{-3.6}=\frac{0.96}{-3.6}$$

Dividing by -3.6 undoes the multiplication by -3.6.

$$p^{2}+\frac{3.6}{-3.6}p=\frac{0.96}{-3.6}$$

Divide 3.6 by -3.6 by multiplying 3.6 by the reciprocal of -3.6.

$$p^{2}-p=\frac{0.96}{-3.6}$$

Divide 0.96 by -3.6 by multiplying 0.96 by the reciprocal of -3.6.

$$p^{2}-p=-\frac{4}{15}$$

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.

$$p^{2}-p+\left(-\frac{1}{2}\right)^{2}=-\frac{4}{15}+\left(-\frac{1}{2}\right)^{2}$$

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

$$p^{2}-p+\frac{1}{4}=-\frac{4}{15}+\frac{1}{4}$$

Add -\frac{4}{15} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}-p+\frac{1}{4}=-\frac{1}{60}$$

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}.

$$\left(p-\frac{1}{2}\right)^{2}=-\frac{1}{60}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{1}{60}}$$

Simplify.

$$p-\frac{1}{2}=\frac{\sqrt{15}i}{30} p-\frac{1}{2}=-\frac{\sqrt{15}i}{30}$$

Add \frac{1}{2} to both sides of the equation.

$$p=\frac{\sqrt{15}i}{30}+\frac{1}{2} p=-\frac{\sqrt{15}i}{30}+\frac{1}{2}$$