Solve left(4x-40right)left(500-40xright)=8000

$$\left( 4x-40 \right) \left( 500-40x \right) = 8000$$

Solve for x (complex solution)

$$x=\frac{45+5\sqrt{31}i}{4}\approx 11.25+6.959705454i$$

$$x=\frac{-5\sqrt{31}i+45}{4}\approx 11.25-6.959705454i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x-40 by 500-40x and combine like terms.

$$3600x-160x^{2}-20000=8000$$

Subtract 8000 from both sides.

$$3600x-160x^{2}-20000-8000=0$$

Subtract 8000 from -20000 to get -28000.

$$3600x-160x^{2}-28000=0$$

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.

$$-160x^{2}+3600x-28000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -160 for a, 3600 for b, and -28000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3600±\sqrt{3600^{2}-4\left(-160\right)\left(-28000\right)}}{2\left(-160\right)}$$

Square 3600.

$$x=\frac{-3600±\sqrt{12960000-4\left(-160\right)\left(-28000\right)}}{2\left(-160\right)}$$

Multiply -4 times -160.

$$x=\frac{-3600±\sqrt{12960000+640\left(-28000\right)}}{2\left(-160\right)}$$

Multiply 640 times -28000.

$$x=\frac{-3600±\sqrt{12960000-17920000}}{2\left(-160\right)}$$

Add 12960000 to -17920000.

$$x=\frac{-3600±\sqrt{-4960000}}{2\left(-160\right)}$$

Take the square root of -4960000.

$$x=\frac{-3600±400\sqrt{31}i}{2\left(-160\right)}$$

Multiply 2 times -160.

$$x=\frac{-3600±400\sqrt{31}i}{-320}$$

Now solve the equation x=\frac{-3600±400\sqrt{31}i}{-320} when ± is plus. Add -3600 to 400i\sqrt{31}.

$$x=\frac{-3600+400\sqrt{31}i}{-320}$$

Divide -3600+400i\sqrt{31} by -320.

$$x=\frac{-5\sqrt{31}i+45}{4}$$

Now solve the equation x=\frac{-3600±400\sqrt{31}i}{-320} when ± is minus. Subtract 400i\sqrt{31} from -3600.

$$x=\frac{-400\sqrt{31}i-3600}{-320}$$

Divide -3600-400i\sqrt{31} by -320.

$$x=\frac{45+5\sqrt{31}i}{4}$$

The equation is now solved.

$$x=\frac{-5\sqrt{31}i+45}{4} x=\frac{45+5\sqrt{31}i}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 4x-40 by 500-40x and combine like terms.

$$3600x-160x^{2}-20000=8000$$

Add 20000 to both sides.

$$3600x-160x^{2}=8000+20000$$

Add 8000 and 20000 to get 28000.

$$3600x-160x^{2}=28000$$

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.

$$-160x^{2}+3600x=28000$$

Divide both sides by -160.

$$\frac{-160x^{2}+3600x}{-160}=\frac{28000}{-160}$$

Dividing by -160 undoes the multiplication by -160.

$$x^{2}+\frac{3600}{-160}x=\frac{28000}{-160}$$

Reduce the fraction \frac{3600}{-160} to lowest terms by extracting and canceling out 80.

$$x^{2}-\frac{45}{2}x=\frac{28000}{-160}$$

Divide 28000 by -160.

$$x^{2}-\frac{45}{2}x=-175$$

Divide -\frac{45}{2}, the coefficient of the x term, by 2 to get -\frac{45}{4}. Then add the square of -\frac{45}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{45}{2}x+\left(-\frac{45}{4}\right)^{2}=-175+\left(-\frac{45}{4}\right)^{2}$$

Square -\frac{45}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{45}{2}x+\frac{2025}{16}=-175+\frac{2025}{16}$$

Add -175 to \frac{2025}{16}.

$$x^{2}-\frac{45}{2}x+\frac{2025}{16}=-\frac{775}{16}$$

Factor x^{2}-\frac{45}{2}x+\frac{2025}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{45}{4}\right)^{2}=-\frac{775}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{45}{4}\right)^{2}}=\sqrt{-\frac{775}{16}}$$

Simplify.

$$x-\frac{45}{4}=\frac{5\sqrt{31}i}{4} x-\frac{45}{4}=-\frac{5\sqrt{31}i}{4}$$

Add \frac{45}{4} to both sides of the equation.

$$x=\frac{45+5\sqrt{31}i}{4} x=\frac{-5\sqrt{31}i+45}{4}$$