Solve -500000x^2+45x-9*10^-6=0

$$-500000 { x }^{ 2 } +45x-9 \times { 10 }^{ -6 } = 0$$

Solve for x

$$x=\frac{3\sqrt{223}}{1000000}+\frac{9}{200000}\approx 0.0000898$$

$$x=-\frac{3\sqrt{223}}{1000000}+\frac{9}{200000}\approx 0.0000002$$

Steps Using the Quadratic Formula

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

$$-500000x^{2}+45x-9\times \frac{1}{1000000}=0$$

Multiply 9 and \frac{1}{1000000} to get \frac{9}{1000000}.

$$-500000x^{2}+45x-\frac{9}{1000000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -500000 for a, 45 for b, and -\frac{9}{1000000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-45±\sqrt{45^{2}-4\left(-500000\right)\left(-\frac{9}{1000000}\right)}}{2\left(-500000\right)}$$

Square 45.

$$x=\frac{-45±\sqrt{2025-4\left(-500000\right)\left(-\frac{9}{1000000}\right)}}{2\left(-500000\right)}$$

Multiply -4 times -500000.

$$x=\frac{-45±\sqrt{2025+2000000\left(-\frac{9}{1000000}\right)}}{2\left(-500000\right)}$$

Multiply 2000000 times -\frac{9}{1000000}.

$$x=\frac{-45±\sqrt{2025-18}}{2\left(-500000\right)}$$

Add 2025 to -18.

$$x=\frac{-45±\sqrt{2007}}{2\left(-500000\right)}$$

Take the square root of 2007.

$$x=\frac{-45±3\sqrt{223}}{2\left(-500000\right)}$$

Multiply 2 times -500000.

$$x=\frac{-45±3\sqrt{223}}{-1000000}$$

Now solve the equation x=\frac{-45±3\sqrt{223}}{-1000000} when ± is plus. Add -45 to 3\sqrt{223}.

$$x=\frac{3\sqrt{223}-45}{-1000000}$$

Divide -45+3\sqrt{223} by -1000000.

$$x=-\frac{3\sqrt{223}}{1000000}+\frac{9}{200000}$$

Now solve the equation x=\frac{-45±3\sqrt{223}}{-1000000} when ± is minus. Subtract 3\sqrt{223} from -45.

$$x=\frac{-3\sqrt{223}-45}{-1000000}$$

Divide -45-3\sqrt{223} by -1000000.

$$x=\frac{3\sqrt{223}}{1000000}+\frac{9}{200000}$$

The equation is now solved.

$$x=-\frac{3\sqrt{223}}{1000000}+\frac{9}{200000} x=\frac{3\sqrt{223}}{1000000}+\frac{9}{200000}$$

Steps for Completing the Square

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

$$-500000x^{2}+45x-9\times \frac{1}{1000000}=0$$

Multiply 9 and \frac{1}{1000000} to get \frac{9}{1000000}.

$$-500000x^{2}+45x-\frac{9}{1000000}=0$$

Add \frac{9}{1000000} to both sides. Anything plus zero gives itself.

$$-500000x^{2}+45x=\frac{9}{1000000}$$

Divide both sides by -500000.

$$\frac{-500000x^{2}+45x}{-500000}=\frac{\frac{9}{1000000}}{-500000}$$

Dividing by -500000 undoes the multiplication by -500000.

$$x^{2}+\frac{45}{-500000}x=\frac{\frac{9}{1000000}}{-500000}$$

Reduce the fraction \frac{45}{-500000} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{9}{100000}x=\frac{\frac{9}{1000000}}{-500000}$$

Divide \frac{9}{1000000} by -500000.

$$x^{2}-\frac{9}{100000}x=-\frac{9}{500000000000}$$

Divide -\frac{9}{100000}, the coefficient of the x term, by 2 to get -\frac{9}{200000}. Then add the square of -\frac{9}{200000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{100000}x+\left(-\frac{9}{200000}\right)^{2}=-\frac{9}{500000000000}+\left(-\frac{9}{200000}\right)^{2}$$

Square -\frac{9}{200000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{100000}x+\frac{81}{40000000000}=-\frac{9}{500000000000}+\frac{81}{40000000000}$$

Add -\frac{9}{500000000000} to \frac{81}{40000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{100000}x+\frac{81}{40000000000}=\frac{2007}{1000000000000}$$

Factor x^{2}-\frac{9}{100000}x+\frac{81}{40000000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{200000}\right)^{2}=\frac{2007}{1000000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{200000}\right)^{2}}=\sqrt{\frac{2007}{1000000000000}}$$

Simplify.

$$x-\frac{9}{200000}=\frac{3\sqrt{223}}{1000000} x-\frac{9}{200000}=-\frac{3\sqrt{223}}{1000000}$$

Add \frac{9}{200000} to both sides of the equation.

$$x=\frac{3\sqrt{223}}{1000000}+\frac{9}{200000} x=-\frac{3\sqrt{223}}{1000000}+\frac{9}{200000}$$