Quadratic Equations

Definition: An equation with one variable, in which the highest power of the variable is two, is known as a quadratic equation.

Examples:

3x² + 4x + 7 = 0

2x² – 50 = 0

1. Standard Form:

A quadratic equation is written as: ax² + bx + c = 0, where a, b, and c are real numbers, and a ≠ 0.

Example: 4x² + 5x – 6 = 0

2. Roots of a Quadratic Equation:

A quadratic equation always has two values for the variable x. These values are called the roots of the equation.

3. Discriminant (D):

For a quadratic equation ax² + bx + c = 0, the expression D = b² – 4ac is called the discriminant (D). It determines the nature of the roots.

4. Types of Quadratic Equations:

• Affected Quadratic Equation: Contains both the square term and the first power term.

Examples: 4x² + 5x = 0, 7x² – 3x = 0

• Pure Quadratic Equation: Contains only the square term.

Examples: x² = 4, 3x² – 8 = 0

◉ Nature of Roots – Discriminant Analysis

The nature of the roots depends on the value of the discriminant (D).

✅ Key Takeaways:

• A quadratic equation always has two roots.
• The discriminant D = b² – 4ac determines the nature of roots.
• If D > 0, roots are real & distinct.
• If D = 0, roots are real & equal.
• If D < 0, roots are complex (imaginary).

Examples:

• Equation: x² – 4x + 4 = 0

D = (-4)² – 4(1)(4) = 16 – 16 = 0 → Real & Equal roots

• Equation: x² – 3x – 4 = 0

D = (-3)² – 4(1)(-4) = 9 + 16 = 25 (> 0) → Real & Unequal roots

• Equation: x² + 2x + 5 = 0

D = (2)² – 4(1)(5) = 4 – 20 = -16 (< 0) → Complex (Imaginary) roots