Solve 240/x-240/x+20-1=0

$$\frac { 240 } { x } - \frac { 240 } { x + 20 } - 1 = 0$$

Solve for x

$$x=-80$$

$$x=60$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+20\right), the least common multiple of x,x+20.

$$\left(x+20\right)\times 240-x\times 240+x\left(x+20\right)\left(-1\right)=0$$

Use the distributive property to multiply x+20 by 240.

$$240x+4800-x\times 240+x\left(x+20\right)\left(-1\right)=0$$

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

$$240x+4800-x\times 240+\left(x^{2}+20x\right)\left(-1\right)=0$$

Use the distributive property to multiply x^{2}+20x by -1.

$$240x+4800-x\times 240-x^{2}-20x=0$$

Combine 240x and -20x to get 220x.

$$220x+4800-x\times 240-x^{2}=0$$

Multiply -1 and 240 to get -240.

$$220x+4800-240x-x^{2}=0$$

Combine 220x and -240x to get -20x.

$$-20x+4800-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}-20x+4800=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+4800. To find a and b, set up a system to be solved.

$$a+b=-20 ab=-4800=-4800$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4800.

$$1,-4800 2,-2400 3,-1600 4,-1200 5,-960 6,-800 8,-600 10,-480 12,-400 15,-320 16,-300 20,-240 24,-200 25,-192 30,-160 32,-150 40,-120 48,-100 50,-96 60,-80 64,-75$$

Calculate the sum for each pair.

$$1-4800=-4799 2-2400=-2398 3-1600=-1597 4-1200=-1196 5-960=-955 6-800=-794 8-600=-592 10-480=-470 12-400=-388 15-320=-305 16-300=-284 20-240=-220 24-200=-176 25-192=-167 30-160=-130 32-150=-118 40-120=-80 48-100=-52 50-96=-46 60-80=-20 64-75=-11$$

The solution is the pair that gives sum -20.

$$a=60 b=-80$$

Rewrite -x^{2}-20x+4800 as \left(-x^{2}+60x\right)+\left(-80x+4800\right).

$$\left(-x^{2}+60x\right)+\left(-80x+4800\right)$$

Factor out x in the first and 80 in the second group.

$$x\left(-x+60\right)+80\left(-x+60\right)$$

Factor out common term -x+60 by using distributive property.

$$\left(-x+60\right)\left(x+80\right)$$

To find equation solutions, solve -x+60=0 and x+80=0.

$$x=60 x=-80$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+20\right), the least common multiple of x,x+20.

$$\left(x+20\right)\times 240-x\times 240+x\left(x+20\right)\left(-1\right)=0$$

Use the distributive property to multiply x+20 by 240.

$$240x+4800-x\times 240+x\left(x+20\right)\left(-1\right)=0$$

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

$$240x+4800-x\times 240+\left(x^{2}+20x\right)\left(-1\right)=0$$

Use the distributive property to multiply x^{2}+20x by -1.

$$240x+4800-x\times 240-x^{2}-20x=0$$

Combine 240x and -20x to get 220x.

$$220x+4800-x\times 240-x^{2}=0$$

Multiply -1 and 240 to get -240.

$$220x+4800-240x-x^{2}=0$$

Combine 220x and -240x to get -20x.

$$-20x+4800-x^{2}=0$$

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.

$$-x^{2}-20x+4800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -20 for b, and 4800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\left(-1\right)\times 4800}}{2\left(-1\right)}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\left(-1\right)\times 4800}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-20\right)±\sqrt{400+4\times 4800}}{2\left(-1\right)}$$

Multiply 4 times 4800.

$$x=\frac{-\left(-20\right)±\sqrt{400+19200}}{2\left(-1\right)}$$

Add 400 to 19200.

$$x=\frac{-\left(-20\right)±\sqrt{19600}}{2\left(-1\right)}$$

Take the square root of 19600.

$$x=\frac{-\left(-20\right)±140}{2\left(-1\right)}$$

The opposite of -20 is 20.

$$x=\frac{20±140}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{20±140}{-2}$$

Now solve the equation x=\frac{20±140}{-2} when ± is plus. Add 20 to 140.

$$x=\frac{160}{-2}$$

Divide 160 by -2.

$$x=-80$$

Now solve the equation x=\frac{20±140}{-2} when ± is minus. Subtract 140 from 20.

$$x=-\frac{120}{-2}$$

Divide -120 by -2.

$$x=60$$

The equation is now solved.

$$x=-80 x=60$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -20,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+20\right), the least common multiple of x,x+20.

$$\left(x+20\right)\times 240-x\times 240+x\left(x+20\right)\left(-1\right)=0$$

Use the distributive property to multiply x+20 by 240.

$$240x+4800-x\times 240+x\left(x+20\right)\left(-1\right)=0$$

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

$$240x+4800-x\times 240+\left(x^{2}+20x\right)\left(-1\right)=0$$

Use the distributive property to multiply x^{2}+20x by -1.

$$240x+4800-x\times 240-x^{2}-20x=0$$

Combine 240x and -20x to get 220x.

$$220x+4800-x\times 240-x^{2}=0$$

Subtract 4800 from both sides. Anything subtracted from zero gives its negation.

$$220x-x\times 240-x^{2}=-4800$$

Multiply -1 and 240 to get -240.

$$220x-240x-x^{2}=-4800$$

Combine 220x and -240x to get -20x.

$$-20x-x^{2}=-4800$$

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.

$$-x^{2}-20x=-4800$$

Divide both sides by -1.

$$\frac{-x^{2}-20x}{-1}=-\frac{4800}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{20}{-1}\right)x=-\frac{4800}{-1}$$

Divide -20 by -1.

$$x^{2}+20x=-\frac{4800}{-1}$$

Divide -4800 by -1.

$$x^{2}+20x=4800$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+20x+10^{2}=4800+10^{2}$$

Square 10.

$$x^{2}+20x+100=4800+100$$

Add 4800 to 100.

$$x^{2}+20x+100=4900$$

Factor x^{2}+20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+10\right)^{2}=4900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{4900}$$

Simplify.

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

Subtract 10 from both sides of the equation.

$$x=60 x=-80$$