Solve (24-x)divx=xdiv100

$$(24-x) \div x=x \div 100$$

Solve for x

$$x=-120$$

$$x=20$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 100x, the least common multiple of x,100.

$$100\left(24-x\right)=xx$$

Multiply x and x to get x^{2}.

$$100\left(24-x\right)=x^{2}$$

Use the distributive property to multiply 100 by 24-x.

$$2400-100x=x^{2}$$

Subtract x^{2} from both sides.

$$2400-100x-x^{2}=0$$

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

$$-x^{2}-100x+2400=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+2400. To find a and b, set up a system to be solved.

$$a+b=-100 ab=-2400=-2400$$

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 -2400.

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

Calculate the sum for each pair.

$$1-2400=-2399 2-1200=-1198 3-800=-797 4-600=-596 5-480=-475 6-400=-394 8-300=-292 10-240=-230 12-200=-188 15-160=-145 16-150=-134 20-120=-100 24-100=-76 25-96=-71 30-80=-50 32-75=-43 40-60=-20 48-50=-2$$

The solution is the pair that gives sum -100.

$$a=20 b=-120$$

Rewrite -x^{2}-100x+2400 as \left(-x^{2}+20x\right)+\left(-120x+2400\right).

$$\left(-x^{2}+20x\right)+\left(-120x+2400\right)$$

Factor out x in the first and 120 in the second group.

$$x\left(-x+20\right)+120\left(-x+20\right)$$

Factor out common term -x+20 by using distributive property.

$$\left(-x+20\right)\left(x+120\right)$$

To find equation solutions, solve -x+20=0 and x+120=0.

$$x=20 x=-120$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 100x, the least common multiple of x,100.

$$100\left(24-x\right)=xx$$

Multiply x and x to get x^{2}.

$$100\left(24-x\right)=x^{2}$$

Use the distributive property to multiply 100 by 24-x.

$$2400-100x=x^{2}$$

Subtract x^{2} from both sides.

$$2400-100x-x^{2}=0$$

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.

$$-x^{2}-100x+2400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -100 for b, and 2400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4\left(-1\right)\times 2400}}{2\left(-1\right)}$$

Square -100.

$$x=\frac{-\left(-100\right)±\sqrt{10000-4\left(-1\right)\times 2400}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-100\right)±\sqrt{10000+4\times 2400}}{2\left(-1\right)}$$

Multiply 4 times 2400.

$$x=\frac{-\left(-100\right)±\sqrt{10000+9600}}{2\left(-1\right)}$$

Add 10000 to 9600.

$$x=\frac{-\left(-100\right)±\sqrt{19600}}{2\left(-1\right)}$$

Take the square root of 19600.

$$x=\frac{-\left(-100\right)±140}{2\left(-1\right)}$$

The opposite of -100 is 100.

$$x=\frac{100±140}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{100±140}{-2}$$

Now solve the equation x=\frac{100±140}{-2} when ± is plus. Add 100 to 140.

$$x=\frac{240}{-2}$$

Divide 240 by -2.

$$x=-120$$

Now solve the equation x=\frac{100±140}{-2} when ± is minus. Subtract 140 from 100.

$$x=-\frac{40}{-2}$$

Divide -40 by -2.

$$x=20$$

The equation is now solved.

$$x=-120 x=20$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 100x, the least common multiple of x,100.

$$100\left(24-x\right)=xx$$

Multiply x and x to get x^{2}.

$$100\left(24-x\right)=x^{2}$$

Use the distributive property to multiply 100 by 24-x.

$$2400-100x=x^{2}$$

Subtract x^{2} from both sides.

$$2400-100x-x^{2}=0$$

Subtract 2400 from both sides. Anything subtracted from zero gives its negation.

$$-100x-x^{2}=-2400$$

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.

$$-x^{2}-100x=-2400$$

Divide both sides by -1.

$$\frac{-x^{2}-100x}{-1}=-\frac{2400}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{100}{-1}\right)x=-\frac{2400}{-1}$$

Divide -100 by -1.

$$x^{2}+100x=-\frac{2400}{-1}$$

Divide -2400 by -1.

$$x^{2}+100x=2400$$

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.

$$x^{2}+100x+50^{2}=2400+50^{2}$$

Square 50.

$$x^{2}+100x+2500=2400+2500$$

Add 2400 to 2500.

$$x^{2}+100x+2500=4900$$

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}.

$$\left(x+50\right)^{2}=4900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+50\right)^{2}}=\sqrt{4900}$$

Simplify.

$$x+50=70 x+50=-70$$

Subtract 50 from both sides of the equation.

$$x=20 x=-120$$