Solve left(40-xright)left(20+20xright)=1200

$$\left( 40-x \right) \left( 20+20x \right) = 1200$$

Solve for x

$$x = \frac{\sqrt{1441} + 39}{2} \approx 38.480252896$$

$$x=\frac{39-\sqrt{1441}}{2}\approx 0.519747104$$

Steps Using the Quadratic Formula

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

$$800+780x-20x^{2}=1200$$

Subtract 1200 from both sides.

$$800+780x-20x^{2}-1200=0$$

Subtract 1200 from 800 to get -400.

$$-400+780x-20x^{2}=0$$

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.

$$-20x^{2}+780x-400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 780 for b, and -400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-780±\sqrt{780^{2}-4\left(-20\right)\left(-400\right)}}{2\left(-20\right)}$$

Square 780.

$$x=\frac{-780±\sqrt{608400-4\left(-20\right)\left(-400\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-780±\sqrt{608400+80\left(-400\right)}}{2\left(-20\right)}$$

Multiply 80 times -400.

$$x=\frac{-780±\sqrt{608400-32000}}{2\left(-20\right)}$$

Add 608400 to -32000.

$$x=\frac{-780±\sqrt{576400}}{2\left(-20\right)}$$

Take the square root of 576400.

$$x=\frac{-780±20\sqrt{1441}}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-780±20\sqrt{1441}}{-40}$$

Now solve the equation x=\frac{-780±20\sqrt{1441}}{-40} when ± is plus. Add -780 to 20\sqrt{1441}.

$$x=\frac{20\sqrt{1441}-780}{-40}$$

Divide -780+20\sqrt{1441} by -40.

$$x=\frac{39-\sqrt{1441}}{2}$$

Now solve the equation x=\frac{-780±20\sqrt{1441}}{-40} when ± is minus. Subtract 20\sqrt{1441} from -780.

$$x=\frac{-20\sqrt{1441}-780}{-40}$$

Divide -780-20\sqrt{1441} by -40.

$$x=\frac{\sqrt{1441}+39}{2}$$

The equation is now solved.

$$x=\frac{39-\sqrt{1441}}{2} x=\frac{\sqrt{1441}+39}{2}$$

Steps for Completing the Square

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

$$800+780x-20x^{2}=1200$$

Subtract 800 from both sides.

$$780x-20x^{2}=1200-800$$

Subtract 800 from 1200 to get 400.

$$780x-20x^{2}=400$$

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.

$$-20x^{2}+780x=400$$

Divide both sides by -20.

$$\frac{-20x^{2}+780x}{-20}=\frac{400}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{780}{-20}x=\frac{400}{-20}$$

Divide 780 by -20.

$$x^{2}-39x=\frac{400}{-20}$$

Divide 400 by -20.

$$x^{2}-39x=-20$$

Divide -39, the coefficient of the x term, by 2 to get -\frac{39}{2}. Then add the square of -\frac{39}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-39x+\left(-\frac{39}{2}\right)^{2}=-20+\left(-\frac{39}{2}\right)^{2}$$

Square -\frac{39}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-39x+\frac{1521}{4}=-20+\frac{1521}{4}$$

Add -20 to \frac{1521}{4}.

$$x^{2}-39x+\frac{1521}{4}=\frac{1441}{4}$$

Factor x^{2}-39x+\frac{1521}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{39}{2}\right)^{2}=\frac{1441}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{39}{2}\right)^{2}}=\sqrt{\frac{1441}{4}}$$

Simplify.

$$x-\frac{39}{2}=\frac{\sqrt{1441}}{2} x-\frac{39}{2}=-\frac{\sqrt{1441}}{2}$$

Add \frac{39}{2} to both sides of the equation.

$$x=\frac{\sqrt{1441}+39}{2} x=\frac{39-\sqrt{1441}}{2}$$