Solve x^2+3.2*10^-8x-3.2*10^-9=0

$${ x }^{ 2 } +3.2 \times { 10 }^{ -8 } x-3.2 \times { 10 }^{ -9 } =0$$

Solve for x

$$x=\frac{9\sqrt{154321}-1}{62500000}\approx 0.000056553$$

$$x=\frac{-9\sqrt{154321}-1}{62500000}\approx -0.000056585$$

Steps Using the Quadratic Formula

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$x^{2}+3.2\times \frac{1}{100000000}x-3.2\times 10^{-9}=0$$

Multiply 3.2 and \frac{1}{100000000} to get \frac{1}{31250000}.

$$x^{2}+\frac{1}{31250000}x-3.2\times 10^{-9}=0$$

Calculate 10 to the power of -9 and get \frac{1}{1000000000}.

$$x^{2}+\frac{1}{31250000}x-3.2\times \frac{1}{1000000000}=0$$

Multiply 3.2 and \frac{1}{1000000000} to get \frac{1}{312500000}.

$$x^{2}+\frac{1}{31250000}x-\frac{1}{312500000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{1}{31250000} for b, and -\frac{1}{312500000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1}{31250000}±\sqrt{\left(\frac{1}{31250000}\right)^{2}-4\left(-\frac{1}{312500000}\right)}}{2}$$

Square \frac{1}{31250000} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{1}{31250000}±\sqrt{\frac{1}{976562500000000}-4\left(-\frac{1}{312500000}\right)}}{2}$$

Multiply -4 times -\frac{1}{312500000}.

$$x=\frac{-\frac{1}{31250000}±\sqrt{\frac{1}{976562500000000}+\frac{1}{78125000}}}{2}$$

Add \frac{1}{976562500000000} to \frac{1}{78125000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{1}{31250000}±\sqrt{\frac{12500001}{976562500000000}}}{2}$$

Take the square root of \frac{12500001}{976562500000000}.

$$x=\frac{-\frac{1}{31250000}±\frac{9\sqrt{154321}}{31250000}}{2}$$

Now solve the equation x=\frac{-\frac{1}{31250000}±\frac{9\sqrt{154321}}{31250000}}{2} when ± is plus. Add -\frac{1}{31250000} to \frac{9\sqrt{154321}}{31250000}.

$$x=\frac{9\sqrt{154321}-1}{2\times 31250000}$$

Divide \frac{-1+9\sqrt{154321}}{31250000} by 2.

$$x=\frac{9\sqrt{154321}-1}{62500000}$$

Now solve the equation x=\frac{-\frac{1}{31250000}±\frac{9\sqrt{154321}}{31250000}}{2} when ± is minus. Subtract \frac{9\sqrt{154321}}{31250000} from -\frac{1}{31250000}.

$$x=\frac{-9\sqrt{154321}-1}{2\times 31250000}$$

Divide \frac{-1-9\sqrt{154321}}{31250000} by 2.

$$x=\frac{-9\sqrt{154321}-1}{62500000}$$

The equation is now solved.

$$x=\frac{9\sqrt{154321}-1}{62500000} x=\frac{-9\sqrt{154321}-1}{62500000}$$

Steps for Completing the Square

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$x^{2}+3.2\times \frac{1}{100000000}x-3.2\times 10^{-9}=0$$

Multiply 3.2 and \frac{1}{100000000} to get \frac{1}{31250000}.

$$x^{2}+\frac{1}{31250000}x-3.2\times 10^{-9}=0$$

Calculate 10 to the power of -9 and get \frac{1}{1000000000}.

$$x^{2}+\frac{1}{31250000}x-3.2\times \frac{1}{1000000000}=0$$

Multiply 3.2 and \frac{1}{1000000000} to get \frac{1}{312500000}.

$$x^{2}+\frac{1}{31250000}x-\frac{1}{312500000}=0$$

Add \frac{1}{312500000} to both sides. Anything plus zero gives itself.

$$x^{2}+\frac{1}{31250000}x=\frac{1}{312500000}$$

Divide \frac{1}{31250000}, the coefficient of the x term, by 2 to get \frac{1}{62500000}. Then add the square of \frac{1}{62500000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{31250000}x+\left(\frac{1}{62500000}\right)^{2}=\frac{1}{312500000}+\left(\frac{1}{62500000}\right)^{2}$$

Square \frac{1}{62500000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{31250000}x+\frac{1}{3906250000000000}=\frac{1}{312500000}+\frac{1}{3906250000000000}$$

Add \frac{1}{312500000} to \frac{1}{3906250000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{31250000}x+\frac{1}{3906250000000000}=\frac{12500001}{3906250000000000}$$

Factor x^{2}+\frac{1}{31250000}x+\frac{1}{3906250000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{62500000}\right)^{2}=\frac{12500001}{3906250000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{62500000}\right)^{2}}=\sqrt{\frac{12500001}{3906250000000000}}$$

Simplify.

$$x+\frac{1}{62500000}=\frac{9\sqrt{154321}}{62500000} x+\frac{1}{62500000}=-\frac{9\sqrt{154321}}{62500000}$$

Subtract \frac{1}{62500000} from both sides of the equation.

$$x=\frac{9\sqrt{154321}-1}{62500000} x=\frac{-9\sqrt{154321}-1}{62500000}$$