Solve 60000/x*-500+2000=-4x

$$\frac{ 60000 }{ x } \times -500+2000=-4x$$

Solve for x

$$x=-3000$$

$$x=2500$$

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 x.

$$60000\left(-500\right)+x\times 2000=-4xx$$

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

$$60000\left(-500\right)+x\times 2000=-4x^{2}$$

Multiply 60000 and -500 to get -30000000.

$$-30000000+x\times 2000=-4x^{2}$$

Add 4x^{2} to both sides.

$$-30000000+x\times 2000+4x^{2}=0$$

Divide both sides by 4.

$$-7500000+500x+x^{2}=0$$

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

$$x^{2}+500x-7500000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-7500000. To find a and b, set up a system to be solved.

$$a+b=500 ab=1\left(-7500000\right)=-7500000$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -7500000.

$$-1,7500000 -2,3750000 -3,2500000 -4,1875000 -5,1500000 -6,1250000 -8,937500 -10,750000 -12,625000 -15,500000 -16,468750 -20,375000 -24,312500 -25,300000 -30,250000 -32,234375 -40,187500 -48,156250 -50,150000 -60,125000 -75,100000 -80,93750 -96,78125 -100,75000 -120,62500 -125,60000 -150,50000 -160,46875 -200,37500 -240,31250 -250,30000 -300,25000 -375,20000 -400,18750 -480,15625 -500,15000 -600,12500 -625,12000 -750,10000 -800,9375 -1000,7500 -1200,6250 -1250,6000 -1500,5000 -1875,4000 -2000,3750 -2400,3125 -2500,3000$$

Calculate the sum for each pair.

$$-1+7500000=7499999 -2+3750000=3749998 -3+2500000=2499997 -4+1875000=1874996 -5+1500000=1499995 -6+1250000=1249994 -8+937500=937492 -10+750000=749990 -12+625000=624988 -15+500000=499985 -16+468750=468734 -20+375000=374980 -24+312500=312476 -25+300000=299975 -30+250000=249970 -32+234375=234343 -40+187500=187460 -48+156250=156202 -50+150000=149950 -60+125000=124940 -75+100000=99925 -80+93750=93670 -96+78125=78029 -100+75000=74900 -120+62500=62380 -125+60000=59875 -150+50000=49850 -160+46875=46715 -200+37500=37300 -240+31250=31010 -250+30000=29750 -300+25000=24700 -375+20000=19625 -400+18750=18350 -480+15625=15145 -500+15000=14500 -600+12500=11900 -625+12000=11375 -750+10000=9250 -800+9375=8575 -1000+7500=6500 -1200+6250=5050 -1250+6000=4750 -1500+5000=3500 -1875+4000=2125 -2000+3750=1750 -2400+3125=725 -2500+3000=500$$

The solution is the pair that gives sum 500.

$$a=-2500 b=3000$$

Rewrite x^{2}+500x-7500000 as \left(x^{2}-2500x\right)+\left(3000x-7500000\right).

$$\left(x^{2}-2500x\right)+\left(3000x-7500000\right)$$

Factor out x in the first and 3000 in the second group.

$$x\left(x-2500\right)+3000\left(x-2500\right)$$

Factor out common term x-2500 by using distributive property.

$$\left(x-2500\right)\left(x+3000\right)$$

To find equation solutions, solve x-2500=0 and x+3000=0.

$$x=2500 x=-3000$$

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 x.

$$60000\left(-500\right)+x\times 2000=-4xx$$

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

$$60000\left(-500\right)+x\times 2000=-4x^{2}$$

Multiply 60000 and -500 to get -30000000.

$$-30000000+x\times 2000=-4x^{2}$$

Add 4x^{2} to both sides.

$$-30000000+x\times 2000+4x^{2}=0$$

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.

$$4x^{2}+2000x-30000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 2000 for b, and -30000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2000±\sqrt{2000^{2}-4\times 4\left(-30000000\right)}}{2\times 4}$$

Square 2000.

$$x=\frac{-2000±\sqrt{4000000-4\times 4\left(-30000000\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-2000±\sqrt{4000000-16\left(-30000000\right)}}{2\times 4}$$

Multiply -16 times -30000000.

$$x=\frac{-2000±\sqrt{4000000+480000000}}{2\times 4}$$

Add 4000000 to 480000000.

$$x=\frac{-2000±\sqrt{484000000}}{2\times 4}$$

Take the square root of 484000000.

$$x=\frac{-2000±22000}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-2000±22000}{8}$$

Now solve the equation x=\frac{-2000±22000}{8} when ± is plus. Add -2000 to 22000.

$$x=\frac{20000}{8}$$

Divide 20000 by 8.

$$x=2500$$

Now solve the equation x=\frac{-2000±22000}{8} when ± is minus. Subtract 22000 from -2000.

$$x=-\frac{24000}{8}$$

Divide -24000 by 8.

$$x=-3000$$

The equation is now solved.

$$x=2500 x=-3000$$

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 x.

$$60000\left(-500\right)+x\times 2000=-4xx$$

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

$$60000\left(-500\right)+x\times 2000=-4x^{2}$$

Multiply 60000 and -500 to get -30000000.

$$-30000000+x\times 2000=-4x^{2}$$

Add 4x^{2} to both sides.

$$-30000000+x\times 2000+4x^{2}=0$$

Add 30000000 to both sides. Anything plus zero gives itself.

$$x\times 2000+4x^{2}=30000000$$

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.

$$4x^{2}+2000x=30000000$$

Divide both sides by 4.

$$\frac{4x^{2}+2000x}{4}=\frac{30000000}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{2000}{4}x=\frac{30000000}{4}$$

Divide 2000 by 4.

$$x^{2}+500x=\frac{30000000}{4}$$

Divide 30000000 by 4.

$$x^{2}+500x=7500000$$

Divide 500, the coefficient of the x term, by 2 to get 250. Then add the square of 250 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+500x+250^{2}=7500000+250^{2}$$

Square 250.

$$x^{2}+500x+62500=7500000+62500$$

Add 7500000 to 62500.

$$x^{2}+500x+62500=7562500$$

Factor x^{2}+500x+62500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+250\right)^{2}=7562500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+250\right)^{2}}=\sqrt{7562500}$$

Simplify.

$$x+250=2750 x+250=-2750$$

Subtract 250 from both sides of the equation.

$$x=2500 x=-3000$$