Solve 300x-15x^2=1800

$$300x-15 { x }^{ 2 } = 1800$$

Solve for x (complex solution)

$$x=10+2\sqrt{5}i\approx 10+4.472135955i$$

$$x=-2\sqrt{5}i+10\approx 10-4.472135955i$$

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.

$$-15x^{2}+300x=1800$$

Subtract 1800 from both sides of the equation.

$$-15x^{2}+300x-1800=1800-1800$$

Subtracting 1800 from itself leaves 0.

$$-15x^{2}+300x-1800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 300 for b, and -1800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-300±\sqrt{300^{2}-4\left(-15\right)\left(-1800\right)}}{2\left(-15\right)}$$

Square 300.

$$x=\frac{-300±\sqrt{90000-4\left(-15\right)\left(-1800\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-300±\sqrt{90000+60\left(-1800\right)}}{2\left(-15\right)}$$

Multiply 60 times -1800.

$$x=\frac{-300±\sqrt{90000-108000}}{2\left(-15\right)}$$

Add 90000 to -108000.

$$x=\frac{-300±\sqrt{-18000}}{2\left(-15\right)}$$

Take the square root of -18000.

$$x=\frac{-300±60\sqrt{5}i}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-300±60\sqrt{5}i}{-30}$$

Now solve the equation x=\frac{-300±60\sqrt{5}i}{-30} when ± is plus. Add -300 to 60i\sqrt{5}.

$$x=\frac{-300+60\sqrt{5}i}{-30}$$

Divide -300+60i\sqrt{5} by -30.

$$x=-2\sqrt{5}i+10$$

Now solve the equation x=\frac{-300±60\sqrt{5}i}{-30} when ± is minus. Subtract 60i\sqrt{5} from -300.

$$x=\frac{-60\sqrt{5}i-300}{-30}$$

Divide -300-60i\sqrt{5} by -30.

$$x=10+2\sqrt{5}i$$

The equation is now solved.

$$x=-2\sqrt{5}i+10 x=10+2\sqrt{5}i$$

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.

$$-15x^{2}+300x=1800$$

Divide both sides by -15.

$$\frac{-15x^{2}+300x}{-15}=\frac{1800}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\frac{300}{-15}x=\frac{1800}{-15}$$

Divide 300 by -15.

$$x^{2}-20x=\frac{1800}{-15}$$

Divide 1800 by -15.

$$x^{2}-20x=-120$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-120+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-120+100$$

Add -120 to 100.

$$x^{2}-20x+100=-20$$

Factor x^{2}-20x+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}.

$$\left(x-10\right)^{2}=-20$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{-20}$$

Simplify.

$$x-10=2\sqrt{5}i x-10=-2\sqrt{5}i$$

Add 10 to both sides of the equation.

$$x=10+2\sqrt{5}i x=-2\sqrt{5}i+10$$