Solve -500000x^2+4.5x-9*10^-6=0

$$-500000 { x }^{ 2 } +4.5x-9 \times { 10 }^{ -6 } = 0$$

Solve for x

$$x=0.000003$$

$$x=0.000006$$

Steps Using the Quadratic Formula

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

$$-500000x^{2}+4.5x-9\times \frac{1}{1000000}=0$$

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

$$-500000x^{2}+4.5x-\frac{9}{1000000}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -500000 for a, 4.5 for b, and -\frac{9}{1000000} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4.5±\sqrt{4.5^{2}-4\left(-500000\right)\left(-\frac{9}{1000000}\right)}}{2\left(-500000\right)}$$

Square 4.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-4.5±\sqrt{20.25-4\left(-500000\right)\left(-\frac{9}{1000000}\right)}}{2\left(-500000\right)}$$

Multiply -4 times -500000.

$$x=\frac{-4.5±\sqrt{20.25+2000000\left(-\frac{9}{1000000}\right)}}{2\left(-500000\right)}$$

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

$$x=\frac{-4.5±\sqrt{20.25-18}}{2\left(-500000\right)}$$

Add 20.25 to -18.

$$x=\frac{-4.5±\sqrt{2.25}}{2\left(-500000\right)}$$

Take the square root of 2.25.

$$x=\frac{-4.5±\frac{3}{2}}{2\left(-500000\right)}$$

Multiply 2 times -500000.

$$x=\frac{-4.5±\frac{3}{2}}{-1000000}$$

Now solve the equation x=\frac{-4.5±\frac{3}{2}}{-1000000} when ± is plus. Add -4.5 to \frac{3}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{3}{-1000000}$$

Divide -3 by -1000000.

$$x=\frac{3}{1000000}$$

Now solve the equation x=\frac{-4.5±\frac{3}{2}}{-1000000} when ± is minus. Subtract \frac{3}{2} from -4.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{6}{-1000000}$$

Reduce the fraction \frac{-6}{-1000000} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{500000}$$

The equation is now solved.

$$x=\frac{3}{1000000} x=\frac{3}{500000}$$

Steps for Completing the Square

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

$$-500000x^{2}+4.5x-9\times \frac{1}{1000000}=0$$

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

$$-500000x^{2}+4.5x-\frac{9}{1000000}=0$$

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

$$-500000x^{2}+4.5x=\frac{9}{1000000}$$

Divide both sides by -500000.

$$\frac{-500000x^{2}+4.5x}{-500000}=\frac{\frac{9}{1000000}}{-500000}$$

Dividing by -500000 undoes the multiplication by -500000.

$$x^{2}+\frac{4.5}{-500000}x=\frac{\frac{9}{1000000}}{-500000}$$

Divide 4.5 by -500000.

$$x^{2}-0.000009x=\frac{\frac{9}{1000000}}{-500000}$$

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

$$x^{2}-0.000009x=-\frac{9}{500000000000}$$

Divide -0.000009, the coefficient of the x term, by 2 to get -0.0000045. Then add the square of -0.0000045 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-0.000009x+\left(-0.0000045\right)^{2}=-\frac{9}{500000000000}+\left(-0.0000045\right)^{2}$$

Square -0.0000045 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-0.000009x+0.00000000002025=-\frac{9}{500000000000}+0.00000000002025$$

Add -\frac{9}{500000000000} to 0.00000000002025 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-0.000009x+0.00000000002025=\frac{9}{4000000000000}$$

Factor x^{2}-0.000009x+0.00000000002025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.0000045\right)^{2}=\frac{9}{4000000000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.0000045\right)^{2}}=\sqrt{\frac{9}{4000000000000}}$$

Simplify.

$$x-0.0000045=\frac{3}{2000000} x-0.0000045=-\frac{3}{2000000}$$

Add 0.0000045 to both sides of the equation.

$$x=\frac{3}{500000} x=\frac{3}{1000000}$$