Solve x^2-(2*10^-7)x-(1*10^-14)=0

$$x ^ { 2 } - ( 2 \times 10 ^ { - 7 } ) x - ( 1 \times 10 ^ { - 14 } ) = 0$$

Solve for x

$$x=\frac{\sqrt{2}+1}{10000000}\approx 0.000000241$$

$$x=\frac{1-\sqrt{2}}{10000000}\approx -0.000000041$$

Steps Using the Quadratic Formula

Calculate 10 to the power of -7 and get \frac{1}{10000000}.

$$x^{2}-2\times \frac{1}{10000000}x-1\times 10^{-14}=0$$

Multiply 2 and \frac{1}{10000000} to get \frac{1}{5000000}.

$$x^{2}-\frac{1}{5000000}x-1\times 10^{-14}=0$$

Calculate 10 to the power of -14 and get \frac{1}{100000000000000}.

$$x^{2}-\frac{1}{5000000}x-1\times \frac{1}{100000000000000}=0$$

Multiply 1 and \frac{1}{100000000000000} to get \frac{1}{100000000000000}.

$$x^{2}-\frac{1}{5000000}x-\frac{1}{100000000000000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -\frac{1}{5000000} for b, and -\frac{1}{100000000000000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{1}{5000000}\right)±\sqrt{\left(-\frac{1}{5000000}\right)^{2}-4\left(-\frac{1}{100000000000000}\right)}}{2}$$

Square -\frac{1}{5000000} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{1}{5000000}\right)±\sqrt{\frac{1}{25000000000000}-4\left(-\frac{1}{100000000000000}\right)}}{2}$$

Multiply -4 times -\frac{1}{100000000000000}.

$$x=\frac{-\left(-\frac{1}{5000000}\right)±\sqrt{\frac{1+1}{25000000000000}}}{2}$$

Add \frac{1}{25000000000000} to \frac{1}{25000000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{1}{5000000}\right)±\sqrt{\frac{1}{12500000000000}}}{2}$$

Take the square root of \frac{1}{12500000000000}.

$$x=\frac{-\left(-\frac{1}{5000000}\right)±\frac{\sqrt{2}}{5000000}}{2}$$

The opposite of -\frac{1}{5000000} is \frac{1}{5000000}.

$$x=\frac{\frac{1}{5000000}±\frac{\sqrt{2}}{5000000}}{2}$$

Now solve the equation x=\frac{\frac{1}{5000000}±\frac{\sqrt{2}}{5000000}}{2} when ± is plus. Add \frac{1}{5000000} to \frac{\sqrt{2}}{5000000}.

$$x=\frac{\sqrt{2}+1}{2\times 5000000}$$

Divide \frac{1+\sqrt{2}}{5000000} by 2.

$$x=\frac{\sqrt{2}+1}{10000000}$$

Now solve the equation x=\frac{\frac{1}{5000000}±\frac{\sqrt{2}}{5000000}}{2} when ± is minus. Subtract \frac{\sqrt{2}}{5000000} from \frac{1}{5000000}.

$$x=\frac{1-\sqrt{2}}{2\times 5000000}$$

Divide \frac{1-\sqrt{2}}{5000000} by 2.

$$x=\frac{1-\sqrt{2}}{10000000}$$

The equation is now solved.

$$x=\frac{\sqrt{2}+1}{10000000} x=\frac{1-\sqrt{2}}{10000000}$$

Steps for Completing the Square

Calculate 10 to the power of -7 and get \frac{1}{10000000}.

$$x^{2}-2\times \frac{1}{10000000}x-1\times 10^{-14}=0$$

Multiply 2 and \frac{1}{10000000} to get \frac{1}{5000000}.

$$x^{2}-\frac{1}{5000000}x-1\times 10^{-14}=0$$

Calculate 10 to the power of -14 and get \frac{1}{100000000000000}.

$$x^{2}-\frac{1}{5000000}x-1\times \frac{1}{100000000000000}=0$$

Multiply 1 and \frac{1}{100000000000000} to get \frac{1}{100000000000000}.

$$x^{2}-\frac{1}{5000000}x-\frac{1}{100000000000000}=0$$

Add \frac{1}{100000000000000} to both sides. Anything plus zero gives itself.

$$x^{2}-\frac{1}{5000000}x=\frac{1}{100000000000000}$$

Divide -\frac{1}{5000000}, the coefficient of the x term, by 2 to get -\frac{1}{10000000}. Then add the square of -\frac{1}{10000000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{5000000}x+\left(-\frac{1}{10000000}\right)^{2}=\frac{1}{100000000000000}+\left(-\frac{1}{10000000}\right)^{2}$$

Square -\frac{1}{10000000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{5000000}x+\frac{1}{100000000000000}=\frac{1+1}{100000000000000}$$

Add \frac{1}{100000000000000} to \frac{1}{100000000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{5000000}x+\frac{1}{100000000000000}=\frac{1}{50000000000000}$$

Factor x^{2}-\frac{1}{5000000}x+\frac{1}{100000000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{10000000}\right)^{2}=\frac{1}{50000000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{10000000}\right)^{2}}=\sqrt{\frac{1}{50000000000000}}$$

Simplify.

$$x-\frac{1}{10000000}=\frac{\sqrt{2}}{10000000} x-\frac{1}{10000000}=-\frac{\sqrt{2}}{10000000}$$

Add \frac{1}{10000000} to both sides of the equation.

$$x=\frac{\sqrt{2}+1}{10000000} x=\frac{1-\sqrt{2}}{10000000}$$