Solve left(240-xright)2left(x+20right)=240x

$$\left( 240-x \right) 2 \left( x+20 \right) = 240x$$

Solve for x

$$x=10\sqrt{73}+50\approx 135.440037453$$

$$x=50-10\sqrt{73}\approx -35.440037453$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 240-x by 2.

$$\left(480-2x\right)\left(x+20\right)=240x$$

Use the distributive property to multiply 480-2x by x+20 and combine like terms.

$$440x+9600-2x^{2}=240x$$

Subtract 240x from both sides.

$$440x+9600-2x^{2}-240x=0$$

Combine 440x and -240x to get 200x.

$$200x+9600-2x^{2}=0$$

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.

$$-2x^{2}+200x+9600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 200 for b, and 9600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-2\right)\times 9600}}{2\left(-2\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-2\right)\times 9600}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-200±\sqrt{40000+8\times 9600}}{2\left(-2\right)}$$

Multiply 8 times 9600.

$$x=\frac{-200±\sqrt{40000+76800}}{2\left(-2\right)}$$

Add 40000 to 76800.

$$x=\frac{-200±\sqrt{116800}}{2\left(-2\right)}$$

Take the square root of 116800.

$$x=\frac{-200±40\sqrt{73}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-200±40\sqrt{73}}{-4}$$

Now solve the equation x=\frac{-200±40\sqrt{73}}{-4} when ± is plus. Add -200 to 40\sqrt{73}.

$$x=\frac{40\sqrt{73}-200}{-4}$$

Divide -200+40\sqrt{73} by -4.

$$x=50-10\sqrt{73}$$

Now solve the equation x=\frac{-200±40\sqrt{73}}{-4} when ± is minus. Subtract 40\sqrt{73} from -200.

$$x=\frac{-40\sqrt{73}-200}{-4}$$

Divide -200-40\sqrt{73} by -4.

$$x=10\sqrt{73}+50$$

The equation is now solved.

$$x=50-10\sqrt{73} x=10\sqrt{73}+50$$

Steps for Completing the Square

Use the distributive property to multiply 240-x by 2.

$$\left(480-2x\right)\left(x+20\right)=240x$$

Use the distributive property to multiply 480-2x by x+20 and combine like terms.

$$440x+9600-2x^{2}=240x$$

Subtract 240x from both sides.

$$440x+9600-2x^{2}-240x=0$$

Combine 440x and -240x to get 200x.

$$200x+9600-2x^{2}=0$$

Subtract 9600 from both sides. Anything subtracted from zero gives its negation.

$$200x-2x^{2}=-9600$$

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.

$$-2x^{2}+200x=-9600$$

Divide both sides by -2.

$$\frac{-2x^{2}+200x}{-2}=-\frac{9600}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{200}{-2}x=-\frac{9600}{-2}$$

Divide 200 by -2.

$$x^{2}-100x=-\frac{9600}{-2}$$

Divide -9600 by -2.

$$x^{2}-100x=4800$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=4800+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=4800+2500$$

Add 4800 to 2500.

$$x^{2}-100x+2500=7300$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=7300$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{7300}$$

Simplify.

$$x-50=10\sqrt{73} x-50=-10\sqrt{73}$$

Add 50 to both sides of the equation.

$$x=10\sqrt{73}+50 x=50-10\sqrt{73}$$