Solve -2(x-40)(x-110)=2250

$$-2(x-40)(x-110)=2250$$

Solve for x

$$x=65$$

$$x=85$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -2 by x-40.

$$\left(-2x+80\right)\left(x-110\right)=2250$$

Use the distributive property to multiply -2x+80 by x-110 and combine like terms.

$$-2x^{2}+300x-8800=2250$$

Subtract 2250 from both sides.

$$-2x^{2}+300x-8800-2250=0$$

Subtract 2250 from -8800 to get -11050.

$$-2x^{2}+300x-11050=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 300 for b, and -11050 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-300±\sqrt{300^{2}-4\left(-2\right)\left(-11050\right)}}{2\left(-2\right)}$$

Square 300.

$$x=\frac{-300±\sqrt{90000-4\left(-2\right)\left(-11050\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-300±\sqrt{90000+8\left(-11050\right)}}{2\left(-2\right)}$$

Multiply 8 times -11050.

$$x=\frac{-300±\sqrt{90000-88400}}{2\left(-2\right)}$$

Add 90000 to -88400.

$$x=\frac{-300±\sqrt{1600}}{2\left(-2\right)}$$

Take the square root of 1600.

$$x=\frac{-300±40}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-300±40}{-4}$$

Now solve the equation x=\frac{-300±40}{-4} when ± is plus. Add -300 to 40.

$$x=-\frac{260}{-4}$$

Divide -260 by -4.

$$x=65$$

Now solve the equation x=\frac{-300±40}{-4} when ± is minus. Subtract 40 from -300.

$$x=-\frac{340}{-4}$$

Divide -340 by -4.

$$x=85$$

The equation is now solved.

$$x=65 x=85$$

Steps for Completing the Square

Use the distributive property to multiply -2 by x-40.

$$\left(-2x+80\right)\left(x-110\right)=2250$$

Use the distributive property to multiply -2x+80 by x-110 and combine like terms.

$$-2x^{2}+300x-8800=2250$$

Add 8800 to both sides.

$$-2x^{2}+300x=2250+8800$$

Add 2250 and 8800 to get 11050.

$$-2x^{2}+300x=11050$$

Divide both sides by -2.

$$\frac{-2x^{2}+300x}{-2}=\frac{11050}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{300}{-2}x=\frac{11050}{-2}$$

Divide 300 by -2.

$$x^{2}-150x=\frac{11050}{-2}$$

Divide 11050 by -2.

$$x^{2}-150x=-5525$$

Divide -150, the coefficient of the x term, by 2 to get -75. Then add the square of -75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-150x+\left(-75\right)^{2}=-5525+\left(-75\right)^{2}$$

Square -75.

$$x^{2}-150x+5625=-5525+5625$$

Add -5525 to 5625.

$$x^{2}-150x+5625=100$$

Factor x^{2}-150x+5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-75\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-75\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-75=10 x-75=-10$$

Add 75 to both sides of the equation.

$$x=85 x=65$$