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 X Mathematics

Chapter 4: Quadratic Equations

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Key Formulas & Equations

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The Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Finds the exact roots of ax² + bx + c = 0.

Discriminant

D = b^2 - 4ac

Determines nature of roots (D > 0: two real roots; D = 0: one real root; D < 0: two complex roots).

Related Formulas from Adjacent Chapters

System Consistency condition

\frac{a_1}{a_2} \neq \frac{b_1}{b_2}

[From Pair of Linear Equations in Two Variables] Condition for a unique solution (intersecting lines).

Infinite Solutions condition

\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}

[From Pair of Linear Equations in Two Variables] Condition for coincident lines.

n-th term of AP

a_n = a + (n-1)d

[From Arithmetic Progressions] Finds the value of the n-th term.