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 XII Mathematics

Chapter 8: Differential Equations

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Syllabus Sections

Key Formulas & Equations

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Linear Differential Equation

\frac{dy}{dx} + P y = Q

Standard form where P and Q are functions of x.

Integrating Factor

IF = e^{\int P \, dx}

Multiplying factor to solve linear differential equations.

Related Formulas from Adjacent Chapters

Area Bounded by Curve

Area = \int_{a}^{b} f(x) \, dx

[From Application of Integrals] Calculates area under curve y = f(x) from x = a to x = b.

Scalar Dot Product

\vec{a} \cdot \vec{b} = |a||b| \cos\theta

[From Vector Algebra] Computes scalar product of two vectors.

Vector Cross Product Magnitude

|\vec{a} \times \vec{b}| = |a||b| \sin\theta

[From Vector Algebra] Magnitude of cross product vector, representing bounded area.