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

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

Solve for x

$$x=80\sqrt{2}+180\approx 293.13708499$$

$$x=180-80\sqrt{2}\approx 66.86291501$$

Steps Using the Quadratic Formula

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

$$130000-1800x+5x^{2}=32000$$

Subtract 32000 from both sides.

$$130000-1800x+5x^{2}-32000=0$$

Subtract 32000 from 130000 to get 98000.

$$98000-1800x+5x^{2}=0$$

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.

$$5x^{2}-1800x+98000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -1800 for b, and 98000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1800\right)±\sqrt{\left(-1800\right)^{2}-4\times 5\times 98000}}{2\times 5}$$

Square -1800.

$$x=\frac{-\left(-1800\right)±\sqrt{3240000-4\times 5\times 98000}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-1800\right)±\sqrt{3240000-20\times 98000}}{2\times 5}$$

Multiply -20 times 98000.

$$x=\frac{-\left(-1800\right)±\sqrt{3240000-1960000}}{2\times 5}$$

Add 3240000 to -1960000.

$$x=\frac{-\left(-1800\right)±\sqrt{1280000}}{2\times 5}$$

Take the square root of 1280000.

$$x=\frac{-\left(-1800\right)±800\sqrt{2}}{2\times 5}$$

The opposite of -1800 is 1800.

$$x=\frac{1800±800\sqrt{2}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{1800±800\sqrt{2}}{10}$$

Now solve the equation x=\frac{1800±800\sqrt{2}}{10} when ± is plus. Add 1800 to 800\sqrt{2}.

$$x=\frac{800\sqrt{2}+1800}{10}$$

Divide 1800+800\sqrt{2} by 10.

$$x=80\sqrt{2}+180$$

Now solve the equation x=\frac{1800±800\sqrt{2}}{10} when ± is minus. Subtract 800\sqrt{2} from 1800.

$$x=\frac{1800-800\sqrt{2}}{10}$$

Divide 1800-800\sqrt{2} by 10.

$$x=180-80\sqrt{2}$$

The equation is now solved.

$$x=80\sqrt{2}+180 x=180-80\sqrt{2}$$

Steps for Completing the Square

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

$$130000-1800x+5x^{2}=32000$$

Subtract 130000 from both sides.

$$-1800x+5x^{2}=32000-130000$$

Subtract 130000 from 32000 to get -98000.

$$-1800x+5x^{2}=-98000$$

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.

$$5x^{2}-1800x=-98000$$

Divide both sides by 5.

$$\frac{5x^{2}-1800x}{5}=-\frac{98000}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\left(-\frac{1800}{5}\right)x=-\frac{98000}{5}$$

Divide -1800 by 5.

$$x^{2}-360x=-\frac{98000}{5}$$

Divide -98000 by 5.

$$x^{2}-360x=-19600$$

Divide -360, the coefficient of the x term, by 2 to get -180. Then add the square of -180 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-360x+\left(-180\right)^{2}=-19600+\left(-180\right)^{2}$$

Square -180.

$$x^{2}-360x+32400=-19600+32400$$

Add -19600 to 32400.

$$x^{2}-360x+32400=12800$$

Factor x^{2}-360x+32400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-180\right)^{2}=12800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-180\right)^{2}}=\sqrt{12800}$$

Simplify.

$$x-180=80\sqrt{2} x-180=-80\sqrt{2}$$

Add 180 to both sides of the equation.

$$x=80\sqrt{2}+180 x=180-80\sqrt{2}$$