Solve (x+100)(1300-5x)=32000

$$( x + 100 ) ( 1300 - 5 x ) = 32000$$

Solve for x

$$x=20\sqrt{65}+80\approx 241.245154966$$

$$x=80-20\sqrt{65}\approx -81.245154966$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x+100 by 1300-5x and combine like terms.

$$800x-5x^{2}+130000=32000$$

Subtract 32000 from both sides.

$$800x-5x^{2}[phone]=0$$

Subtract 32000 from 130000 to get 98000.

$$800x-5x^{2}+98000=0$$

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.

$$-5x^{2}+800x+98000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 800 for b, and 98000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-800±\sqrt{800^{2}-4\left(-5\right)\times 98000}}{2\left(-5\right)}$$

Square 800.

$$x=\frac{-800±\sqrt{640000-4\left(-5\right)\times 98000}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-800±\sqrt{640000+20\times 98000}}{2\left(-5\right)}$$

Multiply 20 times 98000.

$$x=\frac{-800±\sqrt{640000+1960000}}{2\left(-5\right)}$$

Add 640000 to 1960000.

$$x=\frac{-800±\sqrt{2600000}}{2\left(-5\right)}$$

Take the square root of 2600000.

$$x=\frac{-800±200\sqrt{65}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-800±200\sqrt{65}}{-10}$$

Now solve the equation x=\frac{-800±200\sqrt{65}}{-10} when ± is plus. Add -800 to 200\sqrt{65}.

$$x=\frac{200\sqrt{65}-800}{-10}$$

Divide -800+200\sqrt{65} by -10.

$$x=80-20\sqrt{65}$$

Now solve the equation x=\frac{-800±200\sqrt{65}}{-10} when ± is minus. Subtract 200\sqrt{65} from -800.

$$x=\frac{-200\sqrt{65}-800}{-10}$$

Divide -800-200\sqrt{65} by -10.

$$x=20\sqrt{65}+80$$

The equation is now solved.

$$x=80-20\sqrt{65} x=20\sqrt{65}+80$$

Steps for Completing the Square

Use the distributive property to multiply x+100 by 1300-5x and combine like terms.

$$800x-5x^{2}+130000=32000$$

Subtract 130000 from both sides.

$$800x-5x^{2}=32000-130000$$

Subtract 130000 from 32000 to get -98000.

$$800x-5x^{2}=-98000$$

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.

$$-5x^{2}+800x=-98000$$

Divide both sides by -5.

$$\frac{-5x^{2}+800x}{-5}=-\frac{98000}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{800}{-5}x=-\frac{98000}{-5}$$

Divide 800 by -5.

$$x^{2}-160x=-\frac{98000}{-5}$$

Divide -98000 by -5.

$$x^{2}-160x=19600$$

Divide -160, the coefficient of the x term, by 2 to get -80. Then add the square of -80 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-160x+\left(-80\right)^{2}=19600+\left(-80\right)^{2}$$

Square -80.

$$x^{2}-160x+6400=19600+6400$$

Add 19600 to 6400.

$$x^{2}-160x+6400=26000$$

Factor x^{2}-160x+6400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-80\right)^{2}=26000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-80\right)^{2}}=\sqrt{26000}$$

Simplify.

$$x-80=20\sqrt{65} x-80=-20\sqrt{65}$$

Add 80 to both sides of the equation.

$$x=20\sqrt{65}+80 x=80-20\sqrt{65}$$