Solve (10+x)(800-20x)=12000

$$(10+x)(800-20x)=12000$$

Solve for x

$$x=10$$

$$x=20$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10+x by 800-20x and combine like terms.

$$8000+600x-20x^{2}=12000$$

Subtract 12000 from both sides.

$$8000+600x-20x^{2}-12000=0$$

Subtract 12000 from 8000 to get -4000.

$$-4000+600x-20x^{2}=0$$

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.

$$-20x^{2}+600x-4000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 600 for b, and -4000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-600±\sqrt{600^{2}-4\left(-20\right)\left(-4000\right)}}{2\left(-20\right)}$$

Square 600.

$$x=\frac{-600±\sqrt{360000-4\left(-20\right)\left(-4000\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-600±\sqrt{360000+80\left(-4000\right)}}{2\left(-20\right)}$$

Multiply 80 times -4000.

$$x=\frac{-600±\sqrt{360000-320000}}{2\left(-20\right)}$$

Add 360000 to -320000.

$$x=\frac{-600±\sqrt{40000}}{2\left(-20\right)}$$

Take the square root of 40000.

$$x=\frac{-600±200}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-600±200}{-40}$$

Now solve the equation x=\frac{-600±200}{-40} when ± is plus. Add -600 to 200.

$$x=-\frac{400}{-40}$$

Divide -400 by -40.

$$x=10$$

Now solve the equation x=\frac{-600±200}{-40} when ± is minus. Subtract 200 from -600.

$$x=-\frac{800}{-40}$$

Divide -800 by -40.

$$x=20$$

The equation is now solved.

$$x=10 x=20$$

Steps for Completing the Square

Use the distributive property to multiply 10+x by 800-20x and combine like terms.

$$8000+600x-20x^{2}=12000$$

Subtract 8000 from both sides.

$$600x-20x^{2}=12000-8000$$

Subtract 8000 from 12000 to get 4000.

$$600x-20x^{2}=4000$$

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.

$$-20x^{2}+600x=4000$$

Divide both sides by -20.

$$\frac{-20x^{2}+600x}{-20}=\frac{4000}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{600}{-20}x=\frac{4000}{-20}$$

Divide 600 by -20.

$$x^{2}-30x=\frac{4000}{-20}$$

Divide 4000 by -20.

$$x^{2}-30x=-200$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-30x+\left(-15\right)^{2}=-200+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=-200+225$$

Add -200 to 225.

$$x^{2}-30x+225=25$$

Factor x^{2}-30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-15\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-15=5 x-15=-5$$

Add 15 to both sides of the equation.

$$x=20 x=10$$