Quadratic Equations

Definition

A quadratic equation is a second-degree polynomial equation of the form:
ax² + bx + c = 0
where:

• a, b, c are constants,
• a ≠ 0 (if a = 0, it becomes a linear equation),
• x is the variable.

Key Features of Quadratic Equations

• Degree: The highest power of the variable x is 2.
• Parabolic Graph: The graph of a quadratic equation is a parabola that either opens upward (a > 0) or downward (a < 0).
• Roots/Solutions: The values of x that satisfy the equation.

Forms of a Quadratic Equation

1. Standard Form: ax² + bx + c = 0
2. Factored Form: a(x - p)(x - q) = 0, where p and q are the roots.
3. Vertex Form: a(x - h)² + k = 0, where (h, k) is the vertex.

Methods of Solving Quadratic Equations

1. Factoring:
   Solve x² - 5x + 6 = 0:
   Factored form: (x - 2)(x - 3) = 0
   Roots: x = 2, x = 3.
2. Completing the Square:
   Solve x² + 4x - 5 = 0:
   Step 1: x² + 4x = 5
   Step 2: (x + 2)² = 9
   Roots: x = -2 ± 3 (i.e., x = 1, x = -5).
3. Using the Quadratic Formula:

   The quadratic formula is given by:

   x = (-b ± √(b² - 4ac)) / 2a
   Solve 2x² + 3x - 2 = 0:
   a = 2, b = 3, c = -2
x = (-3 ± √(3² - 4(2)(-2))) / (2(2)) = (-3 ± √25) / 4
   Roots: x = 1/2, x = -2.
4. Graphical Method:

   Plot the quadratic function y = ax² + bx + c on a graph. The roots are the x-coordinates where the graph intersects the x-axis.

The Discriminant

The discriminant (Δ) determines the nature of the roots and is given by: Δ = b² - 4ac

• If Δ > 0: Two distinct real roots.
• If Δ = 0: One repeated real root.
• If Δ < 0: Two complex roots.

Applications of Quadratic Equations

• Physics: Projectile motion, free-fall problems.
• Engineering: Design of parabolic structures.
• Economics: Revenue and profit maximization problems.
• Geometry: Finding areas and dimensions.

Examples and Practice Problems

• Solve 3x² - 5x + 2 = 0 using the quadratic formula.
• Determine the nature of the roots of x² + 4x + 5 = 0 using the discriminant.
• If a parabola is given by y = -2x² + 4x + 1, find its vertex and axis of symmetry.