Solve 110x-10x^2=2800

$$110x-10 { x }^{ 2 } =2800$$

Solve for x (complex solution)

$$x=\frac{11+3\sqrt{111}i}{2}\approx 5.5+15.803480629i$$

$$x=\frac{-3\sqrt{111}i+11}{2}\approx 5.5-15.803480629i$$

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.

$$-10x^{2}+110x=2800$$

Subtract 2800 from both sides of the equation.

$$-10x^{2}+110x-2800=2800-2800$$

Subtracting 2800 from itself leaves 0.

$$-10x^{2}+110x-2800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 110 for b, and -2800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-110±\sqrt{110^{2}-4\left(-10\right)\left(-2800\right)}}{2\left(-10\right)}$$

Square 110.

$$x=\frac{-110±\sqrt{12100-4\left(-10\right)\left(-2800\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-110±\sqrt{12100+40\left(-2800\right)}}{2\left(-10\right)}$$

Multiply 40 times -2800.

$$x=\frac{-110±\sqrt{12100-112000}}{2\left(-10\right)}$$

Add 12100 to -112000.

$$x=\frac{-110±\sqrt{-99900}}{2\left(-10\right)}$$

Take the square root of -99900.

$$x=\frac{-110±30\sqrt{111}i}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-110±30\sqrt{111}i}{-20}$$

Now solve the equation x=\frac{-110±30\sqrt{111}i}{-20} when ± is plus. Add -110 to 30i\sqrt{111}.

$$x=\frac{-110+30\sqrt{111}i}{-20}$$

Divide -110+30i\sqrt{111} by -20.

$$x=\frac{-3\sqrt{111}i+11}{2}$$

Now solve the equation x=\frac{-110±30\sqrt{111}i}{-20} when ± is minus. Subtract 30i\sqrt{111} from -110.

$$x=\frac{-30\sqrt{111}i-110}{-20}$$

Divide -110-30i\sqrt{111} by -20.

$$x=\frac{11+3\sqrt{111}i}{2}$$

The equation is now solved.

$$x=\frac{-3\sqrt{111}i+11}{2} x=\frac{11+3\sqrt{111}i}{2}$$

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.

$$-10x^{2}+110x=2800$$

Divide both sides by -10.

$$\frac{-10x^{2}+110x}{-10}=\frac{2800}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{110}{-10}x=\frac{2800}{-10}$$

Divide 110 by -10.

$$x^{2}-11x=\frac{2800}{-10}$$

Divide 2800 by -10.

$$x^{2}-11x=-280$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-280+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=-280+\frac{121}{4}$$

Add -280 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=-\frac{999}{4}$$

Factor x^{2}-11x+\frac{121}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{2}\right)^{2}=-\frac{999}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{-\frac{999}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{3\sqrt{111}i}{2} x-\frac{11}{2}=-\frac{3\sqrt{111}i}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=\frac{11+3\sqrt{111}i}{2} x=\frac{-3\sqrt{111}i+11}{2}$$