Quadratic Equations Guide

A quadratic equation is a polynomial equation of the second degree, formulated as:

$ax^2 + bx + c = 0$

• $a$: leading coefficient
• $b$: linear coefficient
• $c$: constant term

The Quadratic Formula

The quadratic formula is used to determine the values of $x$:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Discriminant $\Delta = b^2 - 4ac$:
  • Defines the number and nature of roots.

Derivation of the Quadratic Formula

1. Begin with: $ax^2 + bx + c = 0$.
2. Rearrange to: $ax^2 + bx = -c$.
3. Normalise: $x^2 + \frac{b}{a}x = -\frac{c}{a}$.
4. Complete the square:
   • Add and subtract $\left(\frac{b}{2a}\right)^2$: $x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 - \frac{c}{a}$
5. Simplify: $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$
6. Solve for $x$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Solving Quadratic Equations: Methods

Completing the Square

This method transforms quadratic equations into the form $(x - h)^2 = k$.

Step-by-Step Transformation Process:

1. Begin with $ax^2 + bx + c = 0$.
2. Normalise: Divide by $a$ if $a \neq 1$.
3. Complete the square on $x^2 + \frac{b}{a}x$.
4. Add and subtract $\left(\frac{b}{2a}\right)^2$.
5. Rewrite in the form $(x - h)^2 = k$.

Example:

• Equation: $x^2 + 6x + 5 = 0$.
• Process:
  • $x^2 + 6x = -5$.
  • Add and subtract $3^2 = 9$:
    • $x^2 + 6x + 9 = -5 + 9$
    • $(x + 3)^2 = 4$
    • $x + 3 = \pm 2$
    • $x = -3 \pm 2$
    • $x = -1$ or $x = -5$

Vertex and Intercepts

Vertex: The point where the parabola changes direction.

Axis of Symmetry: A vertical line through the vertex that divides the parabola equally.

Finding Vertex:

• Vertex Formula: $x = -\frac{b}{2a}$

Example:

• Equation: $y = x^2 - 2x + 1$.
• For vertex, find $x = -\frac{-2}{2(1)} = 1$
• Substitute to find $y = 1^2 - 2(1) + 1 = 0$
• Vertex at $(1, 0)$.

Graph Interpretation

Standard vs. Vertex Form:

• Standard Form: $y = ax^2 + bx + c$
• Vertex Form: $y = a(x-h)^2 + k$

Conversion Example:

• From: $y = 2x^2 + 4x + 1$
• To:
  • $y = 2(x^2 + 2x) + 1$
  • $y = 2(x^2 + 2x + 1 - 1) + 1$
  • $y = 2(x+1)^2 - 2 + 1$
  • $y = 2(x+1)^2 - 1$

Discriminant

Determining Nature of Roots:

• Positive $\Delta$: Two unique roots.
• Zero $\Delta$: One repeated root.
• Negative $\Delta$: Complex roots.

Common Errors Section

• Arithmetic Mistakes: Misuse of signs.
• Parenthesis Misplacement: Leads to incorrect simplifications.

Ensure equations remain balanced and meticulously verify every step.

Practical Applications

The quadratic formula is crucial in fields like physics and economics.

• Projectile Motion: Identifies when a projectile will land.
• Economics: Examines break-even points in profit assessments.

Exam Tips:

• Evaluate the discriminant before solving.
• Utilise graphing calculators to confirm solutions.