Question 1065334: Solve for the indicated letter.
R=vB^2+mB, for B
B=___
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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)
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