Solve x*(1-x)=4/15

$$x \cdot ( 1 - x ) = \frac { 4 } { 15 }$$

Solve for x (complex solution)

$$x=\frac{\sqrt{15}i}{30}+\frac{1}{2}\approx 0.5+0.129099445i$$

$$x=-\frac{\sqrt{15}i}{30}+\frac{1}{2}\approx 0.5-0.129099445i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 1-x.

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

Subtract \frac{4}{15} from both sides.

$$x-x^{2}-\frac{4}{15}=0$$

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.

$$-x^{2}+x-\frac{4}{15}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and -\frac{4}{15} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\left(-\frac{4}{15}\right)}}{2\left(-1\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-1\right)\left(-\frac{4}{15}\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-1±\sqrt{1+4\left(-\frac{4}{15}\right)}}{2\left(-1\right)}$$

Multiply 4 times -\frac{4}{15}.

$$x=\frac{-1±\sqrt{1-\frac{16}{15}}}{2\left(-1\right)}$$

Add 1 to -\frac{16}{15}.

$$x=\frac{-1±\sqrt{-\frac{1}{15}}}{2\left(-1\right)}$$

Take the square root of -\frac{1}{15}.

$$x=\frac{-1±\frac{\sqrt{15}i}{15}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1±\frac{\sqrt{15}i}{15}}{-2}$$

Now solve the equation x=\frac{-1±\frac{\sqrt{15}i}{15}}{-2} when ± is plus. Add -1 to \frac{i\sqrt{15}}{15}.

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

Divide -1+\frac{i\sqrt{15}}{15} by -2.

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

Now solve the equation x=\frac{-1±\frac{\sqrt{15}i}{15}}{-2} when ± is minus. Subtract \frac{i\sqrt{15}}{15} from -1.

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

Divide -1-\frac{i\sqrt{15}}{15} by -2.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply x by 1-x.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 1 by -1.

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

Divide \frac{4}{15} by -1.

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

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.

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

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

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

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.

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

Factor x^{2}-x+\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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

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

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