Solve (20-x)(100+20x)=2240

$$(20-x)(100+20x)=2240$$

Solve for x

$$x = \frac{\sqrt{177} + 15}{2} \approx 14.152067348$$

$$x=\frac{15-\sqrt{177}}{2}\approx 0.847932652$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-x by 100+20x and combine like terms.

$$2000+300x-20x^{2}=2240$$

Subtract 2240 from both sides.

$$2000+300x-20x^{2}-2240=0$$

Subtract 2240 from 2000 to get -240.

$$-240+300x-20x^{2}=0$$

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.

$$-20x^{2}+300x-240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 300 for b, and -240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-300±\sqrt{300^{2}-4\left(-20\right)\left(-240\right)}}{2\left(-20\right)}$$

Square 300.

$$x=\frac{-300±\sqrt{90000-4\left(-20\right)\left(-240\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-300±\sqrt{90000+80\left(-240\right)}}{2\left(-20\right)}$$

Multiply 80 times -240.

$$x=\frac{-300±\sqrt{90000-19200}}{2\left(-20\right)}$$

Add 90000 to -19200.

$$x=\frac{-300±\sqrt{70800}}{2\left(-20\right)}$$

Take the square root of 70800.

$$x=\frac{-300±20\sqrt{177}}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-300±20\sqrt{177}}{-40}$$

Now solve the equation x=\frac{-300±20\sqrt{177}}{-40} when ± is plus. Add -300 to 20\sqrt{177}.

$$x=\frac{20\sqrt{177}-300}{-40}$$

Divide -300+20\sqrt{177} by -40.

$$x=\frac{15-\sqrt{177}}{2}$$

Now solve the equation x=\frac{-300±20\sqrt{177}}{-40} when ± is minus. Subtract 20\sqrt{177} from -300.

$$x=\frac{-20\sqrt{177}-300}{-40}$$

Divide -300-20\sqrt{177} by -40.

$$x=\frac{\sqrt{177}+15}{2}$$

The equation is now solved.

$$x=\frac{15-\sqrt{177}}{2} x=\frac{\sqrt{177}+15}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 20-x by 100+20x and combine like terms.

$$2000+300x-20x^{2}=2240$$

Subtract 2000 from both sides.

$$300x-20x^{2}=2240-2000$$

Subtract 2000 from 2240 to get 240.

$$300x-20x^{2}=240$$

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.

$$-20x^{2}+300x=240$$

Divide both sides by -20.

$$\frac{-20x^{2}+300x}{-20}=\frac{240}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{300}{-20}x=\frac{240}{-20}$$

Divide 300 by -20.

$$x^{2}-15x=\frac{240}{-20}$$

Divide 240 by -20.

$$x^{2}-15x=-12$$

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

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-12+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-12+\frac{225}{4}$$

Add -12 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{177}{4}$$

Factor x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{177}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{177}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{\sqrt{177}}{2} x-\frac{15}{2}=-\frac{\sqrt{177}}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=\frac{\sqrt{177}+15}{2} x=\frac{15-\sqrt{177}}{2}$$