Solve 12000=s^2+20s

$$12000 = s ^ { 2 } + 20 s$$

Solve for s

$$s=-120$$

$$s=100$$

Steps Using Factoring

Swap sides so that all variable terms are on the left hand side.

$$s^{2}+20s=12000$$

Subtract 12000 from both sides.

$$s^{2}+20s-12000=0$$

To solve the equation, factor s^{2}+20s-12000 using formula s^{2}+\left(a+b\right)s+ab=\left(s+a\right)\left(s+b\right). To find a and b, set up a system to be solved.

$$a+b=20 ab=-12000$$

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 -12000.

$$-1,12000 -2,6000 -3,4000 -4,3000 -5,2400 -6,2000 -8,1500 -10,1200 -12,1000 -15,800 -16,750 -20,600 -24,500 -25,480 -30,400 -32,375 -40,300 -48,250 -50,240 -60,200 -75,160 -80,150 -96,125 -100,120$$

Calculate the sum for each pair.

$$-1+12000=11999 -2+6000=5998 -3+4000=3997 -4+3000=2996 -5+2400=2395 -6+2000=1994 -8+1500=1492 -10+1200=1190 -12+1000=988 -15+800=785 -16+750=734 -20+600=580 -24+500=476 -25+480=455 -30+400=370 -32+375=343 -40+300=260 -48+250=202 -50+240=190 -60+200=140 -75+160=85 -80+150=70 -96+125=29 -100+120=20$$

The solution is the pair that gives sum 20.

$$a=-100 b=120$$

Rewrite factored expression \left(s+a\right)\left(s+b\right) using the obtained values.

$$\left(s-100\right)\left(s+120\right)$$

To find equation solutions, solve s-100=0 and s+120=0.

$$s=100 s=-120$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$s^{2}+20s=12000$$

Subtract 12000 from both sides.

$$s^{2}+20s-12000=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as s^{2}+as+bs-12000. To find a and b, set up a system to be solved.

$$a+b=20 ab=1\left(-12000\right)=-12000$$

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 -12000.

$$-1,12000 -2,6000 -3,4000 -4,3000 -5,2400 -6,2000 -8,1500 -10,1200 -12,1000 -15,800 -16,750 -20,600 -24,500 -25,480 -30,400 -32,375 -40,300 -48,250 -50,240 -60,200 -75,160 -80,150 -96,125 -100,120$$

Calculate the sum for each pair.

$$-1+12000=11999 -2+6000=5998 -3+4000=3997 -4+3000=2996 -5+2400=2395 -6+2000=1994 -8+1500=1492 -10+1200=1190 -12+1000=988 -15+800=785 -16+750=734 -20+600=580 -24+500=476 -25+480=455 -30+400=370 -32+375=343 -40+300=260 -48+250=202 -50+240=190 -60+200=140 -75+160=85 -80+150=70 -96+125=29 -100+120=20$$

The solution is the pair that gives sum 20.

$$a=-100 b=120$$

Rewrite s^{2}+20s-12000 as \left(s^{2}-100s\right)+\left(120s-12000\right).

$$\left(s^{2}-100s\right)+\left(120s-12000\right)$$

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

$$s\left(s-100\right)+120\left(s-100\right)$$

Factor out common term s-100 by using distributive property.

$$\left(s-100\right)\left(s+120\right)$$

To find equation solutions, solve s-100=0 and s+120=0.

$$s=100 s=-120$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$s^{2}+20s=12000$$

Subtract 12000 from both sides.

$$s^{2}+20s-12000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 20 for b, and -12000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-20±\sqrt{20^{2}-4\left(-12000\right)}}{2}$$

Square 20.

$$s=\frac{-20±\sqrt{400-4\left(-12000\right)}}{2}$$

Multiply -4 times -12000.

$$s=\frac{-20±\sqrt{400+48000}}{2}$$

Add 400 to 48000.

$$s=\frac{-20±\sqrt{48400}}{2}$$

Take the square root of 48400.

$$s=\frac{-20±220}{2}$$

Now solve the equation s=\frac{-20±220}{2} when ± is plus. Add -20 to 220.

$$s=\frac{200}{2}$$

Divide 200 by 2.

$$s=100$$

Now solve the equation s=\frac{-20±220}{2} when ± is minus. Subtract 220 from -20.

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

Divide -240 by 2.

$$s=-120$$

The equation is now solved.

$$s=100 s=-120$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$s^{2}+20s=12000$$

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.

$$s^{2}+20s+10^{2}=12000+10^{2}$$

Square 10.

$$s^{2}+20s+100=12000+100$$

Add 12000 to 100.

$$s^{2}+20s+100=12100$$

Factor s^{2}+20s+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}.

$$\left(s+10\right)^{2}=12100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(s+10\right)^{2}}=\sqrt{12100}$$

Simplify.

$$s+10=110 s+10=-110$$

Subtract 10 from both sides of the equation.

$$s=100 s=-120$$