Solve (40-x)(20+2x)=200

$$(40-x)(20+2x)=200$$

Solve for x

$$x=5\sqrt{21}+15\approx 37.912878475$$

$$x=15-5\sqrt{21}\approx -7.912878475$$

Steps Using the Quadratic Formula

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

$$800+60x-2x^{2}=200$$

Subtract 200 from both sides.

$$800+60x-2x^{2}-200=0$$

Subtract 200 from 800 to get 600.

$$600+60x-2x^{2}=0$$

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.

$$-2x^{2}+60x+600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 60 for b, and 600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-2\right)\times 600}}{2\left(-2\right)}$$

Square 60.

$$x=\frac{-60±\sqrt{3600-4\left(-2\right)\times 600}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-60±\sqrt{3600+8\times 600}}{2\left(-2\right)}$$

Multiply 8 times 600.

$$x=\frac{-60±\sqrt{3600+4800}}{2\left(-2\right)}$$

Add 3600 to 4800.

$$x=\frac{-60±\sqrt{8400}}{2\left(-2\right)}$$

Take the square root of 8400.

$$x=\frac{-60±20\sqrt{21}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-60±20\sqrt{21}}{-4}$$

Now solve the equation x=\frac{-60±20\sqrt{21}}{-4} when ± is plus. Add -60 to 20\sqrt{21}.

$$x=\frac{20\sqrt{21}-60}{-4}$$

Divide -60+20\sqrt{21} by -4.

$$x=15-5\sqrt{21}$$

Now solve the equation x=\frac{-60±20\sqrt{21}}{-4} when ± is minus. Subtract 20\sqrt{21} from -60.

$$x=\frac{-20\sqrt{21}-60}{-4}$$

Divide -60-20\sqrt{21} by -4.

$$x=5\sqrt{21}+15$$

The equation is now solved.

$$x=15-5\sqrt{21} x=5\sqrt{21}+15$$

Steps for Completing the Square

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

$$800+60x-2x^{2}=200$$

Subtract 800 from both sides.

$$60x-2x^{2}=200-800$$

Subtract 800 from 200 to get -600.

$$60x-2x^{2}=-600$$

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.

$$-2x^{2}+60x=-600$$

Divide both sides by -2.

$$\frac{-2x^{2}+60x}{-2}=-\frac{600}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{60}{-2}x=-\frac{600}{-2}$$

Divide 60 by -2.

$$x^{2}-30x=-\frac{600}{-2}$$

Divide -600 by -2.

$$x^{2}-30x=300$$

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.

$$x^{2}-30x+\left(-15\right)^{2}=300+\left(-15\right)^{2}$$

Square -15.

$$x^{2}-30x+225=300+225$$

Add 300 to 225.

$$x^{2}-30x+225=525$$

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}.

$$\left(x-15\right)^{2}=525$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{525}$$

Simplify.

$$x-15=5\sqrt{21} x-15=-5\sqrt{21}$$

Add 15 to both sides of the equation.

$$x=5\sqrt{21}+15 x=15-5\sqrt{21}$$