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

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

Solve for x

$$x=20$$

Steps Using the Quadratic Formula

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

$$60000+700x-5x^{2}-100\left(300+5x\right)=32000$$

Use the distributive property to multiply -100 by 300+5x.

$$60000+700x-5x^{2}-30000-500x=32000$$

Subtract 30000 from 60000 to get 30000.

$$30000+700x-5x^{2}-500x=32000$$

Combine 700x and -500x to get 200x.

$$30000+200x-5x^{2}=32000$$

Subtract 32000 from both sides.

$$30000+200x-5x^{2}-32000=0$$

Subtract 32000 from 30000 to get -2000.

$$-2000+200x-5x^{2}=0$$

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.

$$-5x^{2}+200x-2000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 200 for b, and -2000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-5\right)\left(-2000\right)}}{2\left(-5\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-5\right)\left(-2000\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-200±\sqrt{40000+20\left(-2000\right)}}{2\left(-5\right)}$$

Multiply 20 times -2000.

$$x=\frac{-200±\sqrt{40000-40000}}{2\left(-5\right)}$$

Add 40000 to -40000.

$$x=\frac{-200±\sqrt{0}}{2\left(-5\right)}$$

Take the square root of 0.

$$x=-\frac{200}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=-\frac{200}{-10}$$

Divide -200 by -10.

$$x=20$$

Steps for Completing the Square

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

$$60000+700x-5x^{2}-100\left(300+5x\right)=32000$$

Use the distributive property to multiply -100 by 300+5x.

$$60000+700x-5x^{2}-30000-500x=32000$$

Subtract 30000 from 60000 to get 30000.

$$30000+700x-5x^{2}-500x=32000$$

Combine 700x and -500x to get 200x.

$$30000+200x-5x^{2}=32000$$

Subtract 30000 from both sides.

$$200x-5x^{2}=32000-30000$$

Subtract 30000 from 32000 to get 2000.

$$200x-5x^{2}=2000$$

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.

$$-5x^{2}+200x=2000$$

Divide both sides by -5.

$$\frac{-5x^{2}+200x}{-5}=\frac{2000}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{200}{-5}x=\frac{2000}{-5}$$

Divide 200 by -5.

$$x^{2}-40x=\frac{2000}{-5}$$

Divide 2000 by -5.

$$x^{2}-40x=-400$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=-400+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=-400+400$$

Add -400 to 400.

$$x^{2}-40x+400=0$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-20=0 x-20=0$$

Add 20 to both sides of the equation.

$$x=20 x=20$$

The equation is now solved. Solutions are the same.

$$x=20$$