Solve x^2+5*10^-3x-10^-6=0

$${ x }^{ 2 } +5 \times { 10 }^{ -3 } x- { 10 }^{ -6 } =0$$

Solve for x

$$x=\frac{\sqrt{29}}{2000}-\frac{1}{400}\approx 0.000192582$$

$$x=-\frac{\sqrt{29}}{2000}-\frac{1}{400}\approx -0.005192582$$

Steps Using the Quadratic Formula

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$x^{2}+5\times \frac{1}{1000}x-10^{-6}=0$$

Multiply 5 and \frac{1}{1000} to get \frac{1}{200}.

$$x^{2}+\frac{1}{200}x-10^{-6}=0$$

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$x^{2}+\frac{1}{200}x-\frac{1}{1000000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{1}{200} for b, and -\frac{1}{1000000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1}{200}±\sqrt{\left(\frac{1}{200}\right)^{2}-4\left(-\frac{1}{1000000}\right)}}{2}$$

Square \frac{1}{200} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{1}{200}±\sqrt{\frac{1}{40000}-4\left(-\frac{1}{1000000}\right)}}{2}$$

Multiply -4 times -\frac{1}{1000000}.

$$x=\frac{-\frac{1}{200}±\sqrt{\frac{1}{40000}+\frac{1}{250000}}}{2}$$

Add \frac{1}{40000} to \frac{1}{250000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{1}{200}±\sqrt{\frac{29}{1000000}}}{2}$$

Take the square root of \frac{29}{1000000}.

$$x=\frac{-\frac{1}{200}±\frac{\sqrt{29}}{1000}}{2}$$

Now solve the equation x=\frac{-\frac{1}{200}±\frac{\sqrt{29}}{1000}}{2} when ± is plus. Add -\frac{1}{200} to \frac{\sqrt{29}}{1000}.

$$x=\frac{\frac{\sqrt{29}}{1000}-\frac{1}{200}}{2}$$

Divide -\frac{1}{200}+\frac{\sqrt{29}}{1000} by 2.

$$x=\frac{\sqrt{29}}{2000}-\frac{1}{400}$$

Now solve the equation x=\frac{-\frac{1}{200}±\frac{\sqrt{29}}{1000}}{2} when ± is minus. Subtract \frac{\sqrt{29}}{1000} from -\frac{1}{200}.

$$x=\frac{-\frac{\sqrt{29}}{1000}-\frac{1}{200}}{2}$$

Divide -\frac{1}{200}-\frac{\sqrt{29}}{1000} by 2.

$$x=-\frac{\sqrt{29}}{2000}-\frac{1}{400}$$

The equation is now solved.

$$x=\frac{\sqrt{29}}{2000}-\frac{1}{400} x=-\frac{\sqrt{29}}{2000}-\frac{1}{400}$$

Steps for Completing the Square

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$x^{2}+5\times \frac{1}{1000}x-10^{-6}=0$$

Multiply 5 and \frac{1}{1000} to get \frac{1}{200}.

$$x^{2}+\frac{1}{200}x-10^{-6}=0$$

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$x^{2}+\frac{1}{200}x-\frac{1}{1000000}=0$$

Add \frac{1}{1000000} to both sides. Anything plus zero gives itself.

$$x^{2}+\frac{1}{200}x=\frac{1}{1000000}$$

Divide \frac{1}{200}, the coefficient of the x term, by 2 to get \frac{1}{400}. Then add the square of \frac{1}{400} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{200}x+\left(\frac{1}{400}\right)^{2}=\frac{1}{1000000}+\left(\frac{1}{400}\right)^{2}$$

Square \frac{1}{400} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{200}x+\frac{1}{160000}=\frac{1}{1000000}+\frac{1}{160000}$$

Add \frac{1}{1000000} to \frac{1}{160000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{200}x+\frac{1}{160000}=\frac{29}{4000000}$$

Factor x^{2}+\frac{1}{200}x+\frac{1}{160000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{400}\right)^{2}=\frac{29}{4000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{400}\right)^{2}}=\sqrt{\frac{29}{4000000}}$$

Simplify.

$$x+\frac{1}{400}=\frac{\sqrt{29}}{2000} x+\frac{1}{400}=-\frac{\sqrt{29}}{2000}$$

Subtract \frac{1}{400} from both sides of the equation.

$$x=\frac{\sqrt{29}}{2000}-\frac{1}{400} x=-\frac{\sqrt{29}}{2000}-\frac{1}{400}$$