Solve x*(0.2-x)=2.8*10^-13

$$x \cdot ( 0.2 - x ) = 2.8 \times 10 ^ { - 13 }$$

Solve for x

$$x=\frac{9\sqrt{3086419753}}{5000000}+0.1\approx 0.2$$

$$x=-\frac{9\sqrt{3086419753}}{5000000}+0.1\approx 1.400005112 \cdot 10^{-12}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 0.2-x.

$$0.2x-x^{2}=2.8\times 10^{-13}$$

Calculate 10 to the power of -13 and get \frac{1}{10000000000000}.

$$0.2x-x^{2}=2.8\times \frac{1}{10000000000000}$$

Multiply 2.8 and \frac{1}{10000000000000} to get \frac{7}{25000000000000}.

$$0.2x-x^{2}=\frac{7}{25000000000000}$$

Subtract \frac{7}{25000000000000} from both sides.

$$0.2x-x^{2}-\frac{7}{25000000000000}=0$$

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.

$$-x^{2}+0.2x-\frac{7}{25000000000000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0.2 for b, and -\frac{7}{25000000000000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.2±\sqrt{0.2^{2}-4\left(-1\right)\left(-\frac{7}{25000000000000}\right)}}{2\left(-1\right)}$$

Square 0.2 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.2±\sqrt{0.04-4\left(-1\right)\left(-\frac{7}{25000000000000}\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-0.2±\sqrt{0.04+4\left(-\frac{7}{25000000000000}\right)}}{2\left(-1\right)}$$

Multiply 4 times -\frac{7}{25000000000000}.

$$x=\frac{-0.2±\sqrt{0.04-\frac{7}{6250000000000}}}{2\left(-1\right)}$$

Add 0.04 to -\frac{7}{6250000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.2±\sqrt{\frac{249999999993}{6250000000000}}}{2\left(-1\right)}$$

Take the square root of \frac{249999999993}{6250000000000}.

$$x=\frac{-0.2±\frac{9\sqrt{3086419753}}{2500000}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-0.2±\frac{9\sqrt{3086419753}}{2500000}}{-2}$$

Now solve the equation x=\frac{-0.2±\frac{9\sqrt{3086419753}}{2500000}}{-2} when ± is plus. Add -0.2 to \frac{9\sqrt{3086419753}}{2500000}.

$$x=\frac{\frac{9\sqrt{3086419753}}{2500000}-\frac{1}{5}}{-2}$$

Divide -\frac{1}{5}+\frac{9\sqrt{3086419753}}{2500000} by -2.

$$x=-\frac{9\sqrt{3086419753}}{5000000}+\frac{1}{10}$$

Now solve the equation x=\frac{-0.2±\frac{9\sqrt{3086419753}}{2500000}}{-2} when ± is minus. Subtract \frac{9\sqrt{3086419753}}{2500000} from -0.2.

$$x=\frac{-\frac{9\sqrt{3086419753}}{2500000}-\frac{1}{5}}{-2}$$

Divide -\frac{1}{5}-\frac{9\sqrt{3086419753}}{2500000} by -2.

$$x=\frac{9\sqrt{3086419753}}{5000000}+\frac{1}{10}$$

The equation is now solved.

$$x=-\frac{9\sqrt{3086419753}}{5000000}+\frac{1}{10} x=\frac{9\sqrt{3086419753}}{5000000}+\frac{1}{10}$$

Steps for Completing the Square

Use the distributive property to multiply x by 0.2-x.

$$0.2x-x^{2}=2.8\times 10^{-13}$$

Calculate 10 to the power of -13 and get \frac{1}{10000000000000}.

$$0.2x-x^{2}=2.8\times \frac{1}{10000000000000}$$

Multiply 2.8 and \frac{1}{10000000000000} to get \frac{7}{25000000000000}.

$$0.2x-x^{2}=\frac{7}{25000000000000}$$

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.

$$-x^{2}+0.2x=\frac{7}{25000000000000}$$

Divide both sides by -1.

$$\frac{-x^{2}+0.2x}{-1}=\frac{\frac{7}{25000000000000}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{0.2}{-1}x=\frac{\frac{7}{25000000000000}}{-1}$$

Divide 0.2 by -1.

$$x^{2}-0.2x=\frac{\frac{7}{25000000000000}}{-1}$$

Divide \frac{7}{25000000000000} by -1.

$$x^{2}-0.2x=-\frac{7}{25000000000000}$$

Divide -0.2, the coefficient of the x term, by 2 to get -0.1. Then add the square of -0.1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-0.2x+\left(-0.1\right)^{2}=-\frac{7}{25000000000000}+\left(-0.1\right)^{2}$$

Square -0.1 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-0.2x+0.01=-\frac{7}{25000000000000}+0.01$$

Add -\frac{7}{25000000000000} to 0.01 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-0.2x+0.01=\frac{249999999993}{25000000000000}$$

Factor x^{2}-0.2x+0.01. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.1\right)^{2}=\frac{249999999993}{25000000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.1\right)^{2}}=\sqrt{\frac{249999999993}{25000000000000}}$$

Simplify.

$$x-0.1=\frac{9\sqrt{3086419753}}{5000000} x-0.1=-\frac{9\sqrt{3086419753}}{5000000}$$

Add 0.1 to both sides of the equation.

$$x=\frac{9\sqrt{3086419753}}{5000000}+\frac{1}{10} x=-\frac{9\sqrt{3086419753}}{5000000}+\frac{1}{10}$$