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

Oct 29, 2015

$R e = \left(- \frac{1}{3}\right)$ and $\text{ } I m = \pm \frac{\sqrt{20}}{12}$

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

#### Explanation:

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

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

Using the common denominator of $4 x$ gives:

Write as: $\text{ } \frac{3 + 12 {x}^{2}}{4 x} = 2$

Giving:

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

Using $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

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

Thus we have:

$x = \frac{- 8 \pm \sqrt{64 - 144}}{2 a}$

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

$x = \frac{- 8 \pm \sqrt{- 80}}{24}$

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

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

Giving:

$x = \left(- \frac{1}{3} \pm \frac{\sqrt{20}}{12} \text{ } i\right)$

Where $R e = \left(- \frac{1}{3}\right)$ and $\text{ } I m = \pm \frac{\sqrt{20}}{12}$

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