What is x if $\frac{3}{4 x} - 3 \frac{x}{4} = 2$ $3/(4x)-3x/4=2$ ?

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Answer 1 · 2015-10-29T21:17:35

$R e = (- \frac{1}{3})$ $Re = (-1/3)$ and $I m = \pm \frac{\sqrt{20}}{12}$ $" "Im= +-(sqrt(20))/12$

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

Given: $\frac{3}{4 x} - 3 \frac{x}{4} = 2$ $3/(4x) -3 x/4 =2$

write as: $\frac{3}{4 x} + \frac{12 x}{4} = 2$ $3/(4x) +(12x)/4 =2$

Using the common denominator of $4 x$ $4x$ gives:

Write as: $\frac{3 + 12 x^{2}}{4 x} = 2$ $" "(3 +12x^2)/(4x) =2$

Giving:

$12 x^{2} - 8 x + 3 = 0$ $12x^2 -8x + 3 =0$

Using $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $x=(-b+-sqrt(b^2-4ac))/(2a)$

std form: $a x^{2} + b x + c = 0$ $ax^2+bx+c=0$

Thus we have:

$x = \frac{- 8 \pm \sqrt{64 - 144}}{2 a}$ $x= (-8+-sqrt(64-144))/(2a)$

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

$x = \frac{- 8 \pm \sqrt{- 80}}{24}$ $x= (-8+-sqrt(-80))/(24)$

But $80 = 2 \times 4 \times 10$ $80 = 2 times 4 times 10$ and $\sqrt{- 1} = i$ $sqrt(-1) = i$

so $\sqrt{- 80} = \sqrt{- 1 \times 2 \times 4 \times 10} = 2 \sqrt{20} i$ $sqrt(-80) = sqrt( -1 times 2 times 4 times 10) = 2 sqrt(20)" "i$

Giving:

$x = (- \frac{1}{3} \pm \frac{\sqrt{20}}{12} i)$ $x= (-1/3 +- (sqrt(20))/12" " i)$

Where $R e = (- \frac{1}{3})$ $Re = (- 1/3 )$ and $I m = \pm \frac{\sqrt{20}}{12}$ $" "Im= +-(sqrt(20))/12$

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