Solve -2(x-40)(x-80)=2250

$$-2(x-40)(x-80)=2250$$

Solve for x (complex solution)

$$x=60+5\sqrt{29}i\approx 60+26.925824036i$$

$$x=-5\sqrt{29}i+60\approx 60-26.925824036i$$

Steps Using the Quadratic Formula

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

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

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

$$-2x^{2}+240x-6400=2250$$

Subtract 2250 from both sides.

$$-2x^{2}+240x-6400-2250=0$$

Subtract 2250 from -6400 to get -8650.

$$-2x^{2}+240x-8650=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 240 for b, and -8650 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-240±\sqrt{240^{2}-4\left(-2\right)\left(-8650\right)}}{2\left(-2\right)}$$

Square 240.

$$x=\frac{-240±\sqrt{57600-4\left(-2\right)\left(-8650\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-240±\sqrt{57600+8\left(-8650\right)}}{2\left(-2\right)}$$

Multiply 8 times -8650.

$$x=\frac{-240±\sqrt{57600-69200}}{2\left(-2\right)}$$

Add 57600 to -69200.

$$x=\frac{-240±\sqrt{-11600}}{2\left(-2\right)}$$

Take the square root of -11600.

$$x=\frac{-240±20\sqrt{29}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-240±20\sqrt{29}i}{-4}$$

Now solve the equation x=\frac{-240±20\sqrt{29}i}{-4} when ± is plus. Add -240 to 20i\sqrt{29}.

$$x=\frac{-240+20\sqrt{29}i}{-4}$$

Divide -240+20i\sqrt{29} by -4.

$$x=-5\sqrt{29}i+60$$

Now solve the equation x=\frac{-240±20\sqrt{29}i}{-4} when ± is minus. Subtract 20i\sqrt{29} from -240.

$$x=\frac{-20\sqrt{29}i-240}{-4}$$

Divide -240-20i\sqrt{29} by -4.

$$x=60+5\sqrt{29}i$$

The equation is now solved.

$$x=-5\sqrt{29}i+60 x=60+5\sqrt{29}i$$

Steps for Completing the Square

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

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

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

$$-2x^{2}+240x-6400=2250$$

Add 6400 to both sides.

$$-2x^{2}+240x=2250+6400$$

Add 2250 and 6400 to get 8650.

$$-2x^{2}+240x=8650$$

Divide both sides by -2.

$$\frac{-2x^{2}+240x}{-2}=\frac{8650}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{240}{-2}x=\frac{8650}{-2}$$

Divide 240 by -2.

$$x^{2}-120x=\frac{8650}{-2}$$

Divide 8650 by -2.

$$x^{2}-120x=-4325$$

Divide -120, the coefficient of the x term, by 2 to get -60. Then add the square of -60 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-120x+\left(-60\right)^{2}=-4325+\left(-60\right)^{2}$$

Square -60.

$$x^{2}-120x+3600=-4325+3600$$

Add -4325 to 3600.

$$x^{2}-120x+3600=-725$$

Factor x^{2}-120x+3600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-60\right)^{2}=-725$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-60\right)^{2}}=\sqrt{-725}$$

Simplify.

$$x-60=5\sqrt{29}i x-60=-5\sqrt{29}i$$

Add 60 to both sides of the equation.

$$x=60+5\sqrt{29}i x=-5\sqrt{29}i+60$$