Solve 200x-15x^2=1125

$$200 x - 15 x ^ { 2 } = 1125$$

Solve for x (complex solution)

$$x=\frac{20+5\sqrt{11}i}{3}\approx 6.666666667+5.527707984i$$

$$x=\frac{-5\sqrt{11}i+20}{3}\approx 6.666666667-5.527707984i$$

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.

$$-15x^{2}+200x=1125$$

Subtract 1125 from both sides of the equation.

$$-15x^{2}+200x-1125=1125-1125$$

Subtracting 1125 from itself leaves 0.

$$-15x^{2}+200x-1125=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 200 for b, and -1125 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-15\right)\left(-1125\right)}}{2\left(-15\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-15\right)\left(-1125\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-200±\sqrt{40000+60\left(-1125\right)}}{2\left(-15\right)}$$

Multiply 60 times -1125.

$$x=\frac{-200±\sqrt{40000-67500}}{2\left(-15\right)}$$

Add 40000 to -67500.

$$x=\frac{-200±\sqrt{-27500}}{2\left(-15\right)}$$

Take the square root of -27500.

$$x=\frac{-200±50\sqrt{11}i}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-200±50\sqrt{11}i}{-30}$$

Now solve the equation x=\frac{-200±50\sqrt{11}i}{-30} when ± is plus. Add -200 to 50i\sqrt{11}.

$$x=\frac{-200+50\sqrt{11}i}{-30}$$

Divide -200+50i\sqrt{11} by -30.

$$x=\frac{-5\sqrt{11}i+20}{3}$$

Now solve the equation x=\frac{-200±50\sqrt{11}i}{-30} when ± is minus. Subtract 50i\sqrt{11} from -200.

$$x=\frac{-50\sqrt{11}i-200}{-30}$$

Divide -200-50i\sqrt{11} by -30.

$$x=\frac{20+5\sqrt{11}i}{3}$$

The equation is now solved.

$$x=\frac{-5\sqrt{11}i+20}{3} x=\frac{20+5\sqrt{11}i}{3}$$

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.

$$-15x^{2}+200x=1125$$

Divide both sides by -15.

$$\frac{-15x^{2}+200x}{-15}=\frac{1125}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\frac{200}{-15}x=\frac{1125}{-15}$$

Reduce the fraction \frac{200}{-15} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{40}{3}x=\frac{1125}{-15}$$

Divide 1125 by -15.

$$x^{2}-\frac{40}{3}x=-75$$

Divide -\frac{40}{3}, the coefficient of the x term, by 2 to get -\frac{20}{3}. Then add the square of -\frac{20}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{40}{3}x+\left(-\frac{20}{3}\right)^{2}=-75+\left(-\frac{20}{3}\right)^{2}$$

Square -\frac{20}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{40}{3}x+\frac{400}{9}=-75+\frac{400}{9}$$

Add -75 to \frac{400}{9}.

$$x^{2}-\frac{40}{3}x+\frac{400}{9}=-\frac{275}{9}$$

Factor x^{2}-\frac{40}{3}x+\frac{400}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{20}{3}\right)^{2}=-\frac{275}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{20}{3}\right)^{2}}=\sqrt{-\frac{275}{9}}$$

Simplify.

$$x-\frac{20}{3}=\frac{5\sqrt{11}i}{3} x-\frac{20}{3}=-\frac{5\sqrt{11}i}{3}$$

Add \frac{20}{3} to both sides of the equation.

$$x=\frac{20+5\sqrt{11}i}{3} x=\frac{-5\sqrt{11}i+20}{3}$$