Solve (200-20(x-10)(x-8)=640

$$(200-20(x-10)(x-8)=640$$

Solve for x (complex solution)

$$x=9+\sqrt{21}i\approx 9+4.582575695i$$

$$x=-\sqrt{21}i+9\approx 9-4.582575695i$$

Steps Using the Quadratic Formula

Subtract 640 from both sides.

$$200-20\left(x-10\right)\left(x-8\right)-640=0$$

Use the distributive property to multiply -20 by x-10.

$$200+\left(-20x+200\right)\left(x-8\right)-640=0$$

Use the distributive property to multiply -20x+200 by x-8 and combine like terms.

$$200-20x^{2}+360x-1600-640=0$$

Subtract 1600 from 200 to get -1400.

$$-1400-20x^{2}+360x-640=0$$

Subtract 640 from -1400 to get -2040.

$$-2040-20x^{2}+360x=0$$

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.

$$-20x^{2}+360x-2040=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 360 for b, and -2040 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-360±\sqrt{360^{2}-4\left(-20\right)\left(-2040\right)}}{2\left(-20\right)}$$

Square 360.

$$x=\frac{-360±\sqrt{129600-4\left(-20\right)\left(-2040\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-360±\sqrt{129600+80\left(-2040\right)}}{2\left(-20\right)}$$

Multiply 80 times -2040.

$$x=\frac{-360±\sqrt{129600-163200}}{2\left(-20\right)}$$

Add 129600 to -163200.

$$x=\frac{-360±\sqrt{-33600}}{2\left(-20\right)}$$

Take the square root of -33600.

$$x=\frac{-360±40\sqrt{21}i}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-360±40\sqrt{21}i}{-40}$$

Now solve the equation x=\frac{-360±40\sqrt{21}i}{-40} when ± is plus. Add -360 to 40i\sqrt{21}.

$$x=\frac{-360+40\sqrt{21}i}{-40}$$

Divide -360+40i\sqrt{21} by -40.

$$x=-\sqrt{21}i+9$$

Now solve the equation x=\frac{-360±40\sqrt{21}i}{-40} when ± is minus. Subtract 40i\sqrt{21} from -360.

$$x=\frac{-40\sqrt{21}i-360}{-40}$$

Divide -360-40i\sqrt{21} by -40.

$$x=9+\sqrt{21}i$$

The equation is now solved.

$$x=-\sqrt{21}i+9 x=9+\sqrt{21}i$$

Steps for Completing the Square

Multiply -1 and 20 to get -20.

$$200-20\left(x-10\right)\left(x-8\right)=640$$

Use the distributive property to multiply -20 by x-10.

$$200+\left(-20x+200\right)\left(x-8\right)=640$$

Use the distributive property to multiply -20x+200 by x-8 and combine like terms.

$$200-20x^{2}+360x-1600=640$$

Subtract 1600 from 200 to get -1400.

$$-1400-20x^{2}+360x=640$$

Add 1400 to both sides.

$$-20x^{2}+360x=640+1400$$

Add 640 and 1400 to get 2040.

$$-20x^{2}+360x=2040$$

Divide both sides by -20.

$$\frac{-20x^{2}+360x}{-20}=\frac{2040}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{360}{-20}x=\frac{2040}{-20}$$

Divide 360 by -20.

$$x^{2}-18x=\frac{2040}{-20}$$

Divide 2040 by -20.

$$x^{2}-18x=-102$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-18x+\left(-9\right)^{2}=-102+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=-102+81$$

Add -102 to 81.

$$x^{2}-18x+81=-21$$

Factor x^{2}-18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9\right)^{2}=-21$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{-21}$$

Simplify.

$$x-9=\sqrt{21}i x-9=-\sqrt{21}i$$

Add 9 to both sides of the equation.

$$x=9+\sqrt{21}i x=-\sqrt{21}i+9$$