Solve -30x^2+18x-3000=150

$$- 30 x ^ { 2 } + 18 x - 3000 = 150$$

Solve for x (complex solution)

$$x=\frac{3+\sqrt{10491}i}{10}\approx 0.3+10.242558274i$$

$$x=\frac{-\sqrt{10491}i+3}{10}\approx 0.3-10.242558274i$$

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.

$$-30x^{2}+18x-3000=150$$

Subtract 150 from both sides of the equation.

$$-30x^{2}+18x-3000-150=150-150$$

Subtracting 150 from itself leaves 0.

$$-30x^{2}+18x-3000-150=0$$

Subtract 150 from -3000.

$$-30x^{2}+18x-3150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -30 for a, 18 for b, and -3150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-30\right)\left(-3150\right)}}{2\left(-30\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-30\right)\left(-3150\right)}}{2\left(-30\right)}$$

Multiply -4 times -30.

$$x=\frac{-18±\sqrt{324+120\left(-3150\right)}}{2\left(-30\right)}$$

Multiply 120 times -3150.

$$x=\frac{-18±\sqrt{324-378000}}{2\left(-30\right)}$$

Add 324 to -378000.

$$x=\frac{-18±\sqrt{-377676}}{2\left(-30\right)}$$

Take the square root of -377676.

$$x=\frac{-18±6\sqrt{10491}i}{2\left(-30\right)}$$

Multiply 2 times -30.

$$x=\frac{-18±6\sqrt{10491}i}{-60}$$

Now solve the equation x=\frac{-18±6\sqrt{10491}i}{-60} when ± is plus. Add -18 to 6i\sqrt{10491}.

$$x=\frac{-18+6\sqrt{10491}i}{-60}$$

Divide -18+6i\sqrt{10491} by -60.

$$x=\frac{-\sqrt{10491}i+3}{10}$$

Now solve the equation x=\frac{-18±6\sqrt{10491}i}{-60} when ± is minus. Subtract 6i\sqrt{10491} from -18.

$$x=\frac{-6\sqrt{10491}i-18}{-60}$$

Divide -18-6i\sqrt{10491} by -60.

$$x=\frac{3+\sqrt{10491}i}{10}$$

The equation is now solved.

$$x=\frac{-\sqrt{10491}i+3}{10} x=\frac{3+\sqrt{10491}i}{10}$$

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.

$$-30x^{2}+18x-3000=150$$

Add 3000 to both sides of the equation.

$$-30x^{2}+18x-3000-\left(-3000\right)=150-\left(-3000\right)$$

Subtracting -3000 from itself leaves 0.

$$-30x^{2}+18x=150-\left(-3000\right)$$

Subtract -3000 from 150.

$$-30x^{2}+18x=3150$$

Divide both sides by -30.

$$\frac{-30x^{2}+18x}{-30}=\frac{3150}{-30}$$

Dividing by -30 undoes the multiplication by -30.

$$x^{2}+\frac{18}{-30}x=\frac{3150}{-30}$$

Reduce the fraction \frac{18}{-30} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{3}{5}x=\frac{3150}{-30}$$

Divide 3150 by -30.

$$x^{2}-\frac{3}{5}x=-105$$

Divide -\frac{3}{5}, the coefficient of the x term, by 2 to get -\frac{3}{10}. Then add the square of -\frac{3}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{5}x+\left(-\frac{3}{10}\right)^{2}=-105+\left(-\frac{3}{10}\right)^{2}$$

Square -\frac{3}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-105+\frac{9}{100}$$

Add -105 to \frac{9}{100}.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{10491}{100}$$

Factor x^{2}-\frac{3}{5}x+\frac{9}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{10}\right)^{2}=-\frac{10491}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{10}\right)^{2}}=\sqrt{-\frac{10491}{100}}$$

Simplify.

$$x-\frac{3}{10}=\frac{\sqrt{10491}i}{10} x-\frac{3}{10}=-\frac{\sqrt{10491}i}{10}$$

Add \frac{3}{10} to both sides of the equation.

$$x=\frac{3+\sqrt{10491}i}{10} x=\frac{-\sqrt{10491}i+3}{10}$$