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

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

Solve for x

$$x=180$$

Steps Using the Quadratic Formula

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

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

Subtract 32000 from both sides.

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

Subtract 32000 from -130000 to get -162000.

$$1800x-5x^{2}-162000=0$$

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.

$$-5x^{2}+1800x-162000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 1800 for b, and -162000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1800±\sqrt{1800^{2}-4\left(-5\right)\left(-162000\right)}}{2\left(-5\right)}$$

Square 1800.

$$x=\frac{-1800±\sqrt{3240000-4\left(-5\right)\left(-162000\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-1800±\sqrt{3240000+20\left(-162000\right)}}{2\left(-5\right)}$$

Multiply 20 times -162000.

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

Add 3240000 to -3240000.

$$x=\frac{-1800±\sqrt{0}}{2\left(-5\right)}$$

Take the square root of 0.

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

Multiply 2 times -5.

$$x=-\frac{1800}{-10}$$

Divide -1800 by -10.

$$x=180$$

Steps for Completing the Square

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

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

Add 130000 to both sides.

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

Add 32000 and 130000 to get 162000.

$$1800x-5x^{2}=162000$$

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.

$$-5x^{2}+1800x=162000$$

Divide both sides by -5.

$$\frac{-5x^{2}+1800x}{-5}=\frac{162000}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{1800}{-5}x=\frac{162000}{-5}$$

Divide 1800 by -5.

$$x^{2}-360x=\frac{162000}{-5}$$

Divide 162000 by -5.

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

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.

$$x^{2}-360x+\left(-180\right)^{2}=-32400+\left(-180\right)^{2}$$

Square -180.

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

Add -32400 to 32400.

$$x^{2}-360x+32400=0$$

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}.

$$\left(x-180\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-180\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-180=0 x-180=0$$

Add 180 to both sides of the equation.

$$x=180 x=180$$

The equation is now solved. Solutions are the same.

$$x=180$$