Solve (200-x)(300-5x)=32000

$$(200-x)(300-5x)=32000$$

Solve for x

$$x=10\sqrt{113}+130\approx 236.301458127$$

$$x=130-10\sqrt{113}\approx 23.698541873$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 200-x by 300-5x and combine like terms.

$$60000-1300x+5x^{2}=32000$$

Subtract 32000 from both sides.

$$60000-1300x+5x^{2}-32000=0$$

Subtract 32000 from 60000 to get 28000.

$$28000-1300x+5x^{2}=0$$

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.

$$5x^{2}-1300x+28000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -1300 for b, and 28000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1300\right)±\sqrt{\left(-1300\right)^{2}-4\times 5\times 28000}}{2\times 5}$$

Square -1300.

$$x=\frac{-\left(-1300\right)±\sqrt{1690000-4\times 5\times 28000}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-1300\right)±\sqrt{1690000-20\times 28000}}{2\times 5}$$

Multiply -20 times 28000.

$$x=\frac{-\left(-1300\right)±\sqrt{1690000-560000}}{2\times 5}$$

Add 1690000 to -560000.

$$x=\frac{-\left(-1300\right)±\sqrt{1130000}}{2\times 5}$$

Take the square root of 1130000.

$$x=\frac{-\left(-1300\right)±100\sqrt{113}}{2\times 5}$$

The opposite of -1300 is 1300.

$$x=\frac{1300±100\sqrt{113}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{1300±100\sqrt{113}}{10}$$

Now solve the equation x=\frac{1300±100\sqrt{113}}{10} when ± is plus. Add 1300 to 100\sqrt{113}.

$$x=\frac{100\sqrt{113}+1300}{10}$$

Divide 1300+100\sqrt{113} by 10.

$$x=10\sqrt{113}+130$$

Now solve the equation x=\frac{1300±100\sqrt{113}}{10} when ± is minus. Subtract 100\sqrt{113} from 1300.

$$x=\frac{1300-100\sqrt{113}}{10}$$

Divide 1300-100\sqrt{113} by 10.

$$x=130-10\sqrt{113}$$

The equation is now solved.

$$x=10\sqrt{113}+130 x=130-10\sqrt{113}$$

Steps for Completing the Square

Use the distributive property to multiply 200-x by 300-5x and combine like terms.

$$60000-1300x+5x^{2}=32000$$

Subtract 60000 from both sides.

$$-1300x+5x^{2}=32000-60000$$

Subtract 60000 from 32000 to get -28000.

$$-1300x+5x^{2}=-28000$$

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.

$$5x^{2}-1300x=-28000$$

Divide both sides by 5.

$$\frac{5x^{2}-1300x}{5}=-\frac{28000}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\left(-\frac{1300}{5}\right)x=-\frac{28000}{5}$$

Divide -1300 by 5.

$$x^{2}-260x=-\frac{28000}{5}$$

Divide -28000 by 5.

$$x^{2}-260x=-5600$$

Divide -260, the coefficient of the x term, by 2 to get -130. Then add the square of -130 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-260x+\left(-130\right)^{2}=-5600+\left(-130\right)^{2}$$

Square -130.

$$x^{2}-260x+16900=-5600+16900$$

Add -5600 to 16900.

$$x^{2}-260x+16900=11300$$

Factor x^{2}-260x+16900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-130\right)^{2}=11300$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-130\right)^{2}}=\sqrt{11300}$$

Simplify.

$$x-130=10\sqrt{113} x-130=-10\sqrt{113}$$

Add 130 to both sides of the equation.

$$x=10\sqrt{113}+130 x=130-10\sqrt{113}$$