Quadratic Equations Without Linear Term

Introduction

• Definition:
  • Quadratic equations without a linear term: Equations of the form $ax^2 + c = 0$.

Purpose: To analyse the nature of quadratic equations of this kind, focusing on their symmetry and the nature of their roots—whether they are real or complex.

Simplifying Quadratic Equations

Isolating $x^2$

• Objective: Simplify $ax^2 + c = 0$.
  • Step-by-step Instructions:
    • Step 1: Divide the equation by $a$: $x^2 = -\frac{c}{a}$
    • Step 2: Apply the square root to both sides:
      • $x = \pm \sqrt{-\frac{c}{a}}$

Conditions for Root Nature

• Real Roots: These occur if $a$ and $c$ have opposite signs, i.e., $-\frac{c}{a} > 0$.
• Imaginary Roots: These arise if $a$ and $c$ have the same sign, i.e., $-\frac{c}{a} < 0$.

Graph Visuals

• Real Roots: A parabola intersects the x-axis when $-\frac{c}{a} > 0$.

  

• Imaginary Roots: The parabola remains fully above or below the x-axis when $-\frac{c}{a} < 0$.

Graphical Representation

• Symmetry: The graph $y = ax^2 + c$ is symmetric along the y-axis.

  

• Vertex: Positioned at $(0, c)$.

  • Direction & Width: Determined by the value of 'a'.

  

Worked Examples

Example 1

• Equation: Solve $3x^2 + 12 = 0$
  • Solution:
    • Isolate $x^2$: $x^2 = -4$
    • Apply square root: $x = \pm \sqrt{-4} = \pm 2i$ (complex roots)
    • Therefore, $x = 2i$ or $x = -2i$

Example 2

• Equation: Solve $2x^2 - 8 = 0$
  • Solution:
    • Isolate $x^2$: $2x^2 = 8$
    • Divide both sides by 2: $x^2 = 4$
    • Apply square root: $x = \pm \sqrt{4} = \pm 2$ (real roots)
    • Therefore, $x = 2$ or $x = -2$

Applications

Physics: Quadratic Motion

• Equation Setup: $h = h_0 - \frac{1}{2}gt^2$

Geometry: Optimisation

• Setup Example: Maximise garden area $A$ with a given perimeter $P$.
  • Equation: $A = \frac{P}{2}x - x^2$, find the maximum for practical layout insights.