Question

The difference between the roots of the equation \(x^2+2x+4 = 0\) is \(\ldots\) (where \(i = \sqrt{-1}\))

• A. \(2\sqrt{3}\)
• B. \(3\sqrt{2}\)
• C. \(2i\sqrt{3}\)
• D. \(i\sqrt{3}\)

Hint:

Solve with the quadratic formula to get complex roots, then subtract them.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A quadratic with negative discriminant has two complex conjugate roots. The difference of the roots is therefore a purely imaginary number.

Step 2: Key Formula or Approach:
For \(ax^2 + bx + c = 0\), the roots are \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The difference of the two roots is \(\dfrac{\sqrt{b^2-4ac}}{a}\) in magnitude.

Step 3: Detailed Explanation:
Here \(a = 1\), \(b = 2\), \(c = 4\), so the discriminant is
\[ D = 2^2 - 4(1)(4) = 4 - 16 = -12 \]
\[ \sqrt{D} = \sqrt{-12} = 2\sqrt{3}\,i \]
The roots are
\[ x = \frac{-2 \pm 2\sqrt{3}\,i}{2} = -1 \pm i\sqrt{3} \]
Difference of the roots:
\[ (-1 + i\sqrt{3}) - (-1 - i\sqrt{3}) = 2i\sqrt{3} \]
The real part of each root cancels, so only the imaginary part is left. Options (A) and (B) are real numbers, which cannot be the difference of two conjugate complex roots. Option (D) takes only half of the imaginary difference.

Final Answer:
The difference between the roots is \(2i\sqrt{3}\), option (C). \[ \boxed{2i\sqrt{3} \text{ (C)}} \]