Question

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

Answer 1

`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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