Question 1065334
equation is:


R = vB^2 + mB


this is a quadratic equation that can be solved through use of the quadratic formula.


subtract R from both sides of the equation to get:


vB^2 + mB - R = 0


if you let:

x = B and:
a = v and:
b = m and:
c = -R. then you get:


ax^2 + bx + c = 0


that's a quadratic equation in standard form.


the quadratic formula states that:


x = (-b + sqrt(b^2 - 4ac)) / (2a)


or:


x = (-b - sqrt(b^2 - 4ac)) /(2a)


all that's left to do is to replace back.


since x = B, then B must be equal to x.
since a = v, then v must be equal to a.
since b = m, then m must be equal to b.
since c = -R then R must be equal to -c.


you will get:


x = (-b + sqrt(b^2 - 4ac)) / (2a) becomes:


B = (-m + sqrt(m^2 + 4vR)) / (2v)


and you will get:


x = (-b - sqrt(b^2 - 4ac)) /(2a) becomes:


B = (-m - sqrt(m^2 + 4vR)) / (2v)


your solution is that B = (-m + sqrt(m^2 + 4vR)) / (2v) or B = B = (-m - sqrt(m^2 + 4vR)) / (2v)



that should be it, if i did the translations correctly.


how can you check?


the way that i do it is to give values to the variables and then solve an equation that i know the solution to.  
if i get the same answer than i assume i did it correctly.


for example:


assume that B = -4 or B = 2.
this means that B + 4 = 0 or B - 2 = 0.
my factors would then be (B + 4) * (B - 2) = 0
multiplying those factors together gets B^2 + 2B - 8 = 0
adding 8 to both sides of the equation gets B^2 + 2B = 8
multiplying both sides of the equation by 2 gets 2B^2 + 4B = 16
flip the equation around to get 16 = 2B^2 + 4B.


16 = 2B^2 + 4B looks a lot like R = vB^2 + mB when:


R = 16
v = 2
m = 4

you want to solve for B using the equations we derived earlier.


they are:


B = (-m + sqrt(m^2 + 4vR)) / (2v) or B = B = (-m - sqrt(m^2 + 4vR)) / (2v)


replacing m with 4 and v with 2 and R with 16, we get:


B = (-4 + sqrt(16 + 128)) / (4) or B = B = (-4 - sqrt(16 + 128)) / (4)


this makes B = 2 or B = -4


those are the values of B that we started from, so we can assume that the formulas are good.


this confirms that your solution is:


B = (-m + sqrt(m^2 + 4vR)) / (2v) or B = B = (-m - sqrt(m^2 + 4vR)) / (2v)