lightbulb Key Concepts

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1. Quadratic Equations

A quadratic equation in the variable x is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.

2. Roots of a Quadratic Equation

A real number α is called a root if aα² + bα + c = 0. The zeros of the quadratic polynomial ax² + bx + c and the roots of the quadratic equation ax² + bx + c = 0 are the same.

3. Solution Methods

• Factorisation: Splitting the middle term so that the product of the parts equals 'ac'.
• Quadratic Formula: The roots are given by x = [-b ± √(b² - 4ac)] / 2a.

4. Nature of Roots

The expression D = b² - 4ac is called the discriminant.

• If D > 0: Two distinct real roots.
• If D = 0: Two equal real roots (Coincident roots).
• If D < 0: No real roots (Roots are imaginary).