Solve (x-100)(300+5(x-200))=32000

$$( x - 100 ) ( 300 + 5 ( x - 200 ) ) = 32000$$

Solve for x

$$x=20\sqrt{17}+120\approx 202.462112512$$

$$x=120-20\sqrt{17}\approx 37.537887488$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5 by x-200.

$$\left(x-100\right)\left(300+5x-1000\right)=32000$$

Subtract 1000 from 300 to get -700.

$$\left(x-100\right)\left(-700+5x\right)=32000$$

Apply the distributive property by multiplying each term of x-100 by each term of -700+5x.

$$-700x+5x^{2}[phone]x=32000$$

Combine -700x and -500x to get -1200x.

$$-1200x+5x^{2}+70000=32000$$

Subtract 32000 from both sides.

$$-1200x+5x^{2}[phone]=0$$

Subtract 32000 from 70000 to get 38000.

$$-1200x+5x^{2}+38000=0$$

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.

$$5x^{2}-1200x+38000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -1200 for b, and 38000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1200\right)±\sqrt{\left(-1200\right)^{2}-4\times 5\times 38000}}{2\times 5}$$

Square -1200.

$$x=\frac{-\left(-1200\right)±\sqrt{1440000-4\times 5\times 38000}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-1200\right)±\sqrt{1440000-20\times 38000}}{2\times 5}$$

Multiply -20 times 38000.

$$x=\frac{-\left(-1200\right)±\sqrt{1440000-760000}}{2\times 5}$$

Add 1440000 to -760000.

$$x=\frac{-\left(-1200\right)±\sqrt{680000}}{2\times 5}$$

Take the square root of 680000.

$$x=\frac{-\left(-1200\right)±200\sqrt{17}}{2\times 5}$$

The opposite of -1200 is 1200.

$$x=\frac{1200±200\sqrt{17}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{1200±200\sqrt{17}}{10}$$

Now solve the equation x=\frac{1200±200\sqrt{17}}{10} when ± is plus. Add 1200 to 200\sqrt{17}.

$$x=\frac{200\sqrt{17}+1200}{10}$$

Divide 1200+200\sqrt{17} by 10.

$$x=20\sqrt{17}+120$$

Now solve the equation x=\frac{1200±200\sqrt{17}}{10} when ± is minus. Subtract 200\sqrt{17} from 1200.

$$x=\frac{1200-200\sqrt{17}}{10}$$

Divide 1200-200\sqrt{17} by 10.

$$x=120-20\sqrt{17}$$

The equation is now solved.

$$x=20\sqrt{17}+120 x=120-20\sqrt{17}$$

Steps for Completing the Square

Use the distributive property to multiply 5 by x-200.

$$\left(x-100\right)\left(300+5x-1000\right)=32000$$

Subtract 1000 from 300 to get -700.

$$\left(x-100\right)\left(-700+5x\right)=32000$$

Apply the distributive property by multiplying each term of x-100 by each term of -700+5x.

$$-700x+5x^{2}[phone]x=32000$$

Combine -700x and -500x to get -1200x.

$$-1200x+5x^{2}+70000=32000$$

Subtract 70000 from both sides.

$$-1200x+5x^{2}=32000-70000$$

Subtract 70000 from 32000 to get -38000.

$$-1200x+5x^{2}=-38000$$

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.

$$5x^{2}-1200x=-38000$$

Divide both sides by 5.

$$\frac{5x^{2}-1200x}{5}=-\frac{38000}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\left(-\frac{1200}{5}\right)x=-\frac{38000}{5}$$

Divide -1200 by 5.

$$x^{2}-240x=-\frac{38000}{5}$$

Divide -38000 by 5.

$$x^{2}-240x=-7600$$

Divide -240, the coefficient of the x term, by 2 to get -120. Then add the square of -120 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-240x+\left(-120\right)^{2}=-7600+\left(-120\right)^{2}$$

Square -120.

$$x^{2}-240x+14400=-7600+14400$$

Add -7600 to 14400.

$$x^{2}-240x+14400=6800$$

Factor x^{2}-240x+14400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-120\right)^{2}=6800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-120\right)^{2}}=\sqrt{6800}$$

Simplify.

$$x-120=20\sqrt{17} x-120=-20\sqrt{17}$$

Add 120 to both sides of the equation.

$$x=20\sqrt{17}+120 x=120-20\sqrt{17}$$