Solve =(4x-40)(500-4x)=8000

$$= ( 4 x - 40 ) ( 500 - 4 x ) = 8000$$

Solve for x

$$x = \frac{5 \sqrt{449} + 135}{2} \approx 120.474050251$$

$$x = \frac{135 - 5 \sqrt{449}}{2} \approx 14.525949749$$

Steps Using the Quadratic Formula

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

$$2160x-16x^{2}-20000=8000$$

Subtract 8000 from both sides.

$$2160x-16x^{2}-20000-8000=0$$

Subtract 8000 from -20000 to get -28000.

$$2160x-16x^{2}-28000=0$$

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.

$$-16x^{2}+2160x-28000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 2160 for b, and -28000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2160±\sqrt{2160^{2}-4\left(-16\right)\left(-28000\right)}}{2\left(-16\right)}$$

Square 2160.

$$x=\frac{-2160±\sqrt{4665600-4\left(-16\right)\left(-28000\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-2160±\sqrt{4665600+64\left(-28000\right)}}{2\left(-16\right)}$$

Multiply 64 times -28000.

$$x=\frac{-2160±\sqrt{4665600-1792000}}{2\left(-16\right)}$$

Add 4665600 to -1792000.

$$x=\frac{-2160±\sqrt{2873600}}{2\left(-16\right)}$$

Take the square root of 2873600.

$$x=\frac{-2160±80\sqrt{449}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-2160±80\sqrt{449}}{-32}$$

Now solve the equation x=\frac{-2160±80\sqrt{449}}{-32} when ± is plus. Add -2160 to 80\sqrt{449}.

$$x=\frac{80\sqrt{449}-2160}{-32}$$

Divide -2160+80\sqrt{449} by -32.

$$x=\frac{135-5\sqrt{449}}{2}$$

Now solve the equation x=\frac{-2160±80\sqrt{449}}{-32} when ± is minus. Subtract 80\sqrt{449} from -2160.

$$x=\frac{-80\sqrt{449}-2160}{-32}$$

Divide -2160-80\sqrt{449} by -32.

$$x=\frac{5\sqrt{449}+135}{2}$$

The equation is now solved.

$$x=\frac{135-5\sqrt{449}}{2} x=\frac{5\sqrt{449}+135}{2}$$

Steps for Completing the Square

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

$$2160x-16x^{2}-20000=8000$$

Add 20000 to both sides.

$$2160x-16x^{2}=8000+20000$$

Add 8000 and 20000 to get 28000.

$$2160x-16x^{2}=28000$$

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.

$$-16x^{2}+2160x=28000$$

Divide both sides by -16.

$$\frac{-16x^{2}+2160x}{-16}=\frac{28000}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{2160}{-16}x=\frac{28000}{-16}$$

Divide 2160 by -16.

$$x^{2}-135x=\frac{28000}{-16}$$

Divide 28000 by -16.

$$x^{2}-135x=-1750$$

Divide -135, the coefficient of the x term, by 2 to get -\frac{135}{2}. Then add the square of -\frac{135}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-135x+\left(-\frac{135}{2}\right)^{2}=-1750+\left(-\frac{135}{2}\right)^{2}$$

Square -\frac{135}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-135x+\frac{18225}{4}=-1750+\frac{18225}{4}$$

Add -1750 to \frac{18225}{4}.

$$x^{2}-135x+\frac{18225}{4}=\frac{11225}{4}$$

Factor x^{2}-135x+\frac{18225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{135}{2}\right)^{2}=\frac{11225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{135}{2}\right)^{2}}=\sqrt{\frac{11225}{4}}$$

Simplify.

$$x-\frac{135}{2}=\frac{5\sqrt{449}}{2} x-\frac{135}{2}=-\frac{5\sqrt{449}}{2}$$

Add \frac{135}{2} to both sides of the equation.

$$x=\frac{5\sqrt{449}+135}{2} x=\frac{135-5\sqrt{449}}{2}$$