Solve -2x^2+220x-4800=800

$$- 2 x ^ { 2 } + 220 x - 4800 = 800$$

Solve for x

$$x=40$$

$$x=70$$

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.

$$-2x^{2}+220x-4800=800$$

Subtract 800 from both sides of the equation.

$$-2x^{2}+220x-4800-800=800-800$$

Subtracting 800 from itself leaves 0.

$$-2x^{2}+220x-4800-800=0$$

Subtract 800 from -4800.

$$-2x^{2}+220x-5600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 220 for b, and -5600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-220±\sqrt{220^{2}-4\left(-2\right)\left(-5600\right)}}{2\left(-2\right)}$$

Square 220.

$$x=\frac{-220±\sqrt{48400-4\left(-2\right)\left(-5600\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-220±\sqrt{48400+8\left(-5600\right)}}{2\left(-2\right)}$$

Multiply 8 times -5600.

$$x=\frac{-220±\sqrt{48400-44800}}{2\left(-2\right)}$$

Add 48400 to -44800.

$$x=\frac{-220±\sqrt{3600}}{2\left(-2\right)}$$

Take the square root of 3600.

$$x=\frac{-220±60}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-220±60}{-4}$$

Now solve the equation x=\frac{-220±60}{-4} when ± is plus. Add -220 to 60.

$$x=-\frac{160}{-4}$$

Divide -160 by -4.

$$x=40$$

Now solve the equation x=\frac{-220±60}{-4} when ± is minus. Subtract 60 from -220.

$$x=-\frac{280}{-4}$$

Divide -280 by -4.

$$x=70$$

The equation is now solved.

$$x=40 x=70$$

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.

$$-2x^{2}+220x-4800=800$$

Add 4800 to both sides of the equation.

$$-2x^{2}+220x-4800-\left(-4800\right)=800-\left(-4800\right)$$

Subtracting -4800 from itself leaves 0.

$$-2x^{2}+220x=800-\left(-4800\right)$$

Subtract -4800 from 800.

$$-2x^{2}+220x=5600$$

Divide both sides by -2.

$$\frac{-2x^{2}+220x}{-2}=\frac{5600}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{220}{-2}x=\frac{5600}{-2}$$

Divide 220 by -2.

$$x^{2}-110x=\frac{5600}{-2}$$

Divide 5600 by -2.

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

Divide -110, the coefficient of the x term, by 2 to get -55. Then add the square of -55 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-110x+\left(-55\right)^{2}=-2800+\left(-55\right)^{2}$$

Square -55.

$$x^{2}-110x+3025=-2800+3025$$

Add -2800 to 3025.

$$x^{2}-110x+3025=225$$

Factor x^{2}-110x+3025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-55\right)^{2}=225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-55\right)^{2}}=\sqrt{225}$$

Simplify.

$$x-55=15 x-55=-15$$

Add 55 to both sides of the equation.

$$x=70 x=40$$