Solve (40-x)(20+2x)=1000

$$( 40 - x ) ( 20 + 2 x ) = 1000$$

Solve for x

$$x=5\sqrt{5}+15\approx 26.180339887$$

$$x=15-5\sqrt{5}\approx 3.819660113$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40-x by 20+2x and combine like terms.

$$800+60x-2x^{2}=1000$$

Subtract 1000 from both sides.

$$800+60x-2x^{2}-1000=0$$

Subtract 1000 from 800 to get -200.

$$-200+60x-2x^{2}=0$$

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.

$$-2x^{2}+60x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 60 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-2\right)\left(-200\right)}}{2\left(-2\right)}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-2\right)\left(-200\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-60±\sqrt{3600+8\left(-200\right)}}{2\left(-2\right)}$$

Multiply 8 times -200.

$$x=\frac{-60±\sqrt{3600-1600}}{2\left(-2\right)}$$

Add 3600 to -1600.

$$x=\frac{-60±\sqrt{2000}}{2\left(-2\right)}$$

Take the square root of 2000.

$$x=\frac{-60±20\sqrt{5}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-60±20\sqrt{5}}{-4}$$

Now solve the equation x=\frac{-60±20\sqrt{5}}{-4} when ± is plus. Add -60 to 20\sqrt{5}.

$$x=\frac{20\sqrt{5}-60}{-4}$$

Divide -60+20\sqrt{5} by -4.

$$x=15-5\sqrt{5}$$

Now solve the equation x=\frac{-60±20\sqrt{5}}{-4} when ± is minus. Subtract 20\sqrt{5} from -60.

$$x=\frac{-20\sqrt{5}-60}{-4}$$

Divide -60-20\sqrt{5} by -4.

$$x=5\sqrt{5}+15$$

The equation is now solved.

$$x=15-5\sqrt{5} x=5\sqrt{5}+15$$

Steps for Completing the Square

Use the distributive property to multiply 40-x by 20+2x and combine like terms.

$$800+60x-2x^{2}=1000$$

Subtract 800 from both sides.

$$60x-2x^{2}=1000-800$$

Subtract 800 from 1000 to get 200.

$$60x-2x^{2}=200$$

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.

$$-2x^{2}+60x=200$$

Divide both sides by -2.

$$\frac{-2x^{2}+60x}{-2}=\frac{200}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{60}{-2}x=\frac{200}{-2}$$

Divide 60 by -2.

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

Divide 200 by -2.

$$x^{2}-30x=-100$$

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.

$$x^{2}-30x+\left(-15\right)^{2}=-100+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=-100+225$$

Add -100 to 225.

$$x^{2}-30x+225=125$$

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}.

$$\left(x-15\right)^{2}=125$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{125}$$

Simplify.

$$x-15=5\sqrt{5} x-15=-5\sqrt{5}$$

Add 15 to both sides of the equation.

$$x=5\sqrt{5}+15 x=15-5\sqrt{5}$$