Quadratic Equations

A quadratic equation is a second-degree polynomial equation in one variable, typically written in the standard form:

ax² + bx + c = 0

where:

• a, b, and c are constants.
• a ≠ 0 (if a = 0, the equation becomes linear).

Key Features of a Quadratic Equation

1. Degree: The highest power of x is 2.
2. Roots: The solutions of the quadratic equation are called roots. These are the values of x that satisfy ax² + bx + c = 0.
3. Discriminant (D): Determines the nature of roots and is given by:

   D = b² - 4ac

   • If D > 0: Two distinct real roots.
   • If D = 0: Two equal real roots (repeated roots).
   • If D < 0: No real roots (complex roots).

Methods of Solving Quadratic Equations

• Factorization
• Completing the Square
• Using the Quadratic Formula:

  x = (-b ± √(b² - 4ac)) / 2a

• Graphing (to visualize roots)

Example Problems

1. Solving by Factorization

Solve x² - 5x + 6 = 0.

x² - 5x + 6 = 0
x² - 2x - 3x + 6 = 0
x(x - 2) - 3(x - 2) = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3
    

2. Solving by Completing the Square

Solve x² + 6x + 5 = 0.

x² + 6x + 5 = 0
x² + 6x = -5
x² + 6x + 9 = 4
(x + 3)² = 4
x + 3 = ±2
x = -1 or x = -5
    

3. Solving by Using the Quadratic Formula

Solve 2x² - 4x - 6 = 0.

x = (-b ± √(b² - 4ac)) / 2a
x = (4 ± √(16 + 48)) / 4
x = (4 ± √64) / 4
x = (4 ± 8) / 4
x = 3 or x = -1
    

4. Solving When D < 0

Solve x² + 4x + 5 = 0.

D = b² - 4ac = 4² - 4(1)(5) = -4
x = (-4 ± √-4) / 2
x = -2 ± i
x = -2 + i or x = -2 - i
    

5. Word Problem Involving Quadratic Equations

The product of two consecutive integers is 56. Find the integers.

Let the integers be x and x + 1.
x(x + 1) = 56
x² + x - 56 = 0
(x + 8)(x - 7) = 0
x = -8 or x = 7
The integers are -8 and -7 or 7 and 8.
    

Applications of Quadratic Equations

• Physics (projectile motion, acceleration)
• Finance (profit maximization, breaking even)
• Geometry (finding dimensions, areas)
• Engineering (design and analysis)