Solve (60+x-40)(240-20x/5)=6400

$$( 60 + x - 40 ) ( 240 - 20 \frac { x } { 5 } ) = 6400$$

Solve for x

$$x=20$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 5.

$$5\left(60+x-40\right)\left(240-20\times \frac{x}{5}\right)=32000$$

Subtract 40 from 60 to get 20.

$$5\left(20+x\right)\left(240-20\times \frac{x}{5}\right)=32000$$

Cancel out 5, the greatest common factor in 20 and 5.

$$5\left(20+x\right)\left(240-4x\right)=32000$$

Use the distributive property to multiply 5 by 20+x.

$$\left(100+5x\right)\left(240-4x\right)=32000$$

Apply the distributive property by multiplying each term of 100+5x by each term of 240-4x.

$$24000-400x+1200x-20x^{2}=32000$$

Combine -400x and 1200x to get 800x.

$$24000+800x-20x^{2}=32000$$

Subtract 32000 from both sides.

$$24000+800x-20x^{2}-32000=0$$

Subtract 32000 from 24000 to get -8000.

$$-8000+800x-20x^{2}=0$$

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.

$$-20x^{2}+800x-8000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 800 for b, and -8000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-800±\sqrt{800^{2}-4\left(-20\right)\left(-8000\right)}}{2\left(-20\right)}$$

Square 800.

$$x=\frac{-800±\sqrt{640000-4\left(-20\right)\left(-8000\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-800±\sqrt{640000+80\left(-8000\right)}}{2\left(-20\right)}$$

Multiply 80 times -8000.

$$x=\frac{-800±\sqrt{640000-640000}}{2\left(-20\right)}$$

Add 640000 to -640000.

$$x=\frac{-800±\sqrt{0}}{2\left(-20\right)}$$

Take the square root of 0.

$$x=-\frac{800}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=-\frac{800}{-40}$$

Divide -800 by -40.

$$x=20$$

Steps for Completing the Square

Multiply both sides of the equation by 5.

$$5\left(60+x-40\right)\left(240-20\times \frac{x}{5}\right)=32000$$

Subtract 40 from 60 to get 20.

$$5\left(20+x\right)\left(240-20\times \frac{x}{5}\right)=32000$$

Cancel out 5, the greatest common factor in 20 and 5.

$$5\left(20+x\right)\left(240-4x\right)=32000$$

Use the distributive property to multiply 5 by 20+x.

$$\left(100+5x\right)\left(240-4x\right)=32000$$

Apply the distributive property by multiplying each term of 100+5x by each term of 240-4x.

$$24000-400x+1200x-20x^{2}=32000$$

Combine -400x and 1200x to get 800x.

$$24000+800x-20x^{2}=32000$$

Subtract 24000 from both sides.

$$800x-20x^{2}=32000-24000$$

Subtract 24000 from 32000 to get 8000.

$$800x-20x^{2}=8000$$

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.

$$-20x^{2}+800x=8000$$

Divide both sides by -20.

$$\frac{-20x^{2}+800x}{-20}=\frac{8000}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{800}{-20}x=\frac{8000}{-20}$$

Divide 800 by -20.

$$x^{2}-40x=\frac{8000}{-20}$$

Divide 8000 by -20.

$$x^{2}-40x=-400$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=-400+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=-400+400$$

Add -400 to 400.

$$x^{2}-40x+400=0$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-20=0 x-20=0$$

Add 20 to both sides of the equation.

$$x=20 x=20$$

The equation is now solved. Solutions are the same.

$$x=20$$