Expert Answer to "What is x if 34x−3x4=2?"

Marlee Yates

Answered question

2022-02-02

What is x if

\frac{3}{4 x} - 3 \frac{x}{4} = 2

Finn Lucero

Beginner2022-02-03Added 17 answers

R e = (- \frac{1}{3}) and 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 4x gives:
Write as:

\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

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

Giving:

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

Where

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

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