Can a quadratic equation have three roots?

No. According to the Fundamental Theorem of Algebra, a polynomial equation of degree n has exactly n complex roots. A quadratic equation is a second-degree polynomial (n = 2) and thus can have at most two roots.

What are imaginary roots?

When the discriminant D = b² - 4ac is negative, the square root term contains a negative value, yielding roots with the imaginary unit i (where i = √-1). These are complex numbers representing points where the parabola does not cross the x-axis.

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Class XI Mathematics

Chapter 4: Complex Numbers and Quadratic Equations

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Key Formulas & Equations

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Complex Number Polar Form

z = r(\cos\theta + i\sin\theta)

Polar representation using modulus r and argument θ.

Modulus of Complex Number

|z| = \sqrt{x^2 + y^2}

Magnitude of complex number x + iy.

Related Formulas from Adjacent Chapters

Radian Conversion Formula

\text{Radian} = \text{Degree} \times \frac{\pi}{180}

[From Trigonometric Functions] Converts angle from degree measures to radian metrics.

Pythagorean Identity

\sin^2 x + \cos^2 x = 1

[From Trigonometric Functions] Relates squared values of sine and cosine functions.

Inequality sign multiplication

a < b \land c < 0 \Rightarrow ac > bc

[From Linear Inequalities] Reverses inequality sign when multiplying by a negative number.