What is x if #3/(4x)-3x/4=2#?

1 Answer
Oct 29, 2015

Answer:

#Re = (-1/3) # and #" "Im= +-(sqrt(20))/12#

Complex numbers. The plot does not cross the x-axis

Explanation:

Given: #3/(4x) -3 x/4 =2#

write as: #3/(4x) +(12x)/4 =2 #

Using the common denominator of #4x# gives:

Write as: #" "(3 +12x^2)/(4x) =2#

Giving:

#12x^2 -8x + 3 =0#

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

std form: #ax^2+bx+c=0#

Thus we have:

#x= (-8+-sqrt(64-144))/(2a)#

Straight off we have a negative root so the quadratic does not cross the x-axis. Any solution would be expressed as complex numbers.

#x= (-8+-sqrt(-80))/(24)#

But #80 = 2 times 4 times 10# and #sqrt(-1) = i#

so # sqrt(-80) = sqrt( -1 times 2 times 4 times 10) = 2 sqrt(20)" "i#

Giving:

#x= (-1/3 +- (sqrt(20))/12" " i)#

Where #Re = (- 1/3 )# and #" "Im= +-(sqrt(20))/12#

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