Solve (240-x)*2*(x-20)=240x

$$(240-x) \times 2 \times (x-20)=240x$$

Solve for x

$$x=60$$

$$x=80$$

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.

$$520x-9600-2x^{2}=240x$$

Subtract 240x from both sides.

$$520x-9600-2x^{2}-240x=0$$

Combine 520x and -240x to get 280x.

$$280x-9600-2x^{2}=0$$

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.

$$-2x^{2}+280x-9600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 280 for b, and -9600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-280±\sqrt{280^{2}-4\left(-2\right)\left(-9600\right)}}{2\left(-2\right)}$$

Square 280.

$$x=\frac{-280±\sqrt{78400-4\left(-2\right)\left(-9600\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-280±\sqrt{78400+8\left(-9600\right)}}{2\left(-2\right)}$$

Multiply 8 times -9600.

$$x=\frac{-280±\sqrt{78400-76800}}{2\left(-2\right)}$$

Add 78400 to -76800.

$$x=\frac{-280±\sqrt{1600}}{2\left(-2\right)}$$

Take the square root of 1600.

$$x=\frac{-280±40}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-280±40}{-4}$$

Now solve the equation x=\frac{-280±40}{-4} when ± is plus. Add -280 to 40.

$$x=-\frac{240}{-4}$$

Divide -240 by -4.

$$x=60$$

Now solve the equation x=\frac{-280±40}{-4} when ± is minus. Subtract 40 from -280.

$$x=-\frac{320}{-4}$$

Divide -320 by -4.

$$x=80$$

The equation is now solved.

$$x=60 x=80$$

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.

$$520x-9600-2x^{2}=240x$$

Subtract 240x from both sides.

$$520x-9600-2x^{2}-240x=0$$

Combine 520x and -240x to get 280x.

$$280x-9600-2x^{2}=0$$

Add 9600 to both sides. Anything plus zero gives itself.

$$280x-2x^{2}=9600$$

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.

$$-2x^{2}+280x=9600$$

Divide both sides by -2.

$$\frac{-2x^{2}+280x}{-2}=\frac{9600}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{280}{-2}x=\frac{9600}{-2}$$

Divide 280 by -2.

$$x^{2}-140x=\frac{9600}{-2}$$

Divide 9600 by -2.

$$x^{2}-140x=-4800$$

Divide -140, the coefficient of the x term, by 2 to get -70. Then add the square of -70 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-140x+\left(-70\right)^{2}=-4800+\left(-70\right)^{2}$$

Square -70.

$$x^{2}-140x+4900=-4800+4900$$

Add -4800 to 4900.

$$x^{2}-140x+4900=100$$

Factor x^{2}-140x+4900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-70\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-70\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-70=10 x-70=-10$$

Add 70 to both sides of the equation.

$$x=80 x=60$$