Solve 200x^2-9x+120=0

$$200 { x }^{ 2 } -9x+120=0$$

Solve for x (complex solution)

$$x=\frac{9+\sqrt{95919}i}{400}\approx 0.0225+0.774269817i$$

$$x=\frac{-\sqrt{95919}i+9}{400}\approx 0.0225-0.774269817i$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$200x^{2}-9x+120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 200 for a, -9 for b, and 120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 200\times 120}}{2\times 200}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 200\times 120}}{2\times 200}$$

Multiply -4 times 200.

$$x=\frac{-\left(-9\right)±\sqrt{81-800\times 120}}{2\times 200}$$

Multiply -800 times 120.

$$x=\frac{-\left(-9\right)±\sqrt{81-96000}}{2\times 200}$$

Add 81 to -96000.

$$x=\frac{-\left(-9\right)±\sqrt{-95919}}{2\times 200}$$

Take the square root of -95919.

$$x=\frac{-\left(-9\right)±\sqrt{95919}i}{2\times 200}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{95919}i}{2\times 200}$$

Multiply 2 times 200.

$$x=\frac{9±\sqrt{95919}i}{400}$$

Now solve the equation x=\frac{9±\sqrt{95919}i}{400} when ± is plus. Add 9 to i\sqrt{95919}.

$$x=\frac{9+\sqrt{95919}i}{400}$$

Now solve the equation x=\frac{9±\sqrt{95919}i}{400} when ± is minus. Subtract i\sqrt{95919} from 9.

$$x=\frac{-\sqrt{95919}i+9}{400}$$

The equation is now solved.

$$x=\frac{9+\sqrt{95919}i}{400} x=\frac{-\sqrt{95919}i+9}{400}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$200x^{2}-9x+120=0$$

Subtract 120 from both sides of the equation.

$$200x^{2}-9x+120-120=-120$$

Subtracting 120 from itself leaves 0.

$$200x^{2}-9x=-120$$

Divide both sides by 200.

$$\frac{200x^{2}-9x}{200}=-\frac{120}{200}$$

Dividing by 200 undoes the multiplication by 200.

$$x^{2}-\frac{9}{200}x=-\frac{120}{200}$$

Reduce the fraction \frac{-120}{200} to lowest terms by extracting and canceling out 40.

$$x^{2}-\frac{9}{200}x=-\frac{3}{5}$$

Divide -\frac{9}{200}, the coefficient of the x term, by 2 to get -\frac{9}{400}. Then add the square of -\frac{9}{400} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{200}x+\left(-\frac{9}{400}\right)^{2}=-\frac{3}{5}+\left(-\frac{9}{400}\right)^{2}$$

Square -\frac{9}{400} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{200}x+\frac{81}{160000}=-\frac{3}{5}+\frac{81}{160000}$$

Add -\frac{3}{5} to \frac{81}{160000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{200}x+\frac{81}{160000}=-\frac{95919}{160000}$$

Factor x^{2}-\frac{9}{200}x+\frac{81}{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}.

$$\left(x-\frac{9}{400}\right)^{2}=-\frac{95919}{160000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{400}\right)^{2}}=\sqrt{-\frac{95919}{160000}}$$

Simplify.

$$x-\frac{9}{400}=\frac{\sqrt{95919}i}{400} x-\frac{9}{400}=-\frac{\sqrt{95919}i}{400}$$

Add \frac{9}{400} to both sides of the equation.

$$x=\frac{9+\sqrt{95919}i}{400} x=\frac{-\sqrt{95919}i+9}{400}$$