Sample Notes: Quadratics

AS Level Mathematics – Topic 1.1: Quadratics

1. Standard Quadratic Form

• A quadratic expression is written as:
  ax² + bx + c, where:
  • a, b, c are constants
  • a ≠ 0
• This is a second-degree polynomial.

2. Completing the Square

• Used to rewrite a quadratic in the form:
  a(x + p)² + q
• Steps for completing the square for y = ax² + bx + c:
  1. Factor out a from the x² and x terms:
     • y = a(x² + (b/a)x) + c
  2. Add and subtract (b/2a)² inside the bracket:
     • = a[(x + b/2a)² – (b/2a)²] + c
  3. Expand and simplify:
     • = a(x + b/2a)² – ab²/4a² + c
     • = a(x + b/2a)² + (4ac – b²)/4a

Example:

• y = 2x² + 8x + 5
  = 2[(x² + 4x) + 5/2]
  = 2[(x + 2)² – 4 + 5/2]
  = 2(x + 2)² – 3

Applications:

• Easily find vertex: (-b/2a, f(-b/2a))
• Used for graph sketching and maximum/minimum problems

3. Discriminant

• Part of the quadratic formula:
  • b² – 4ac
• Tells the nature of the roots:
  • If b² – 4ac > 0, two distinct real roots
  • If b² – 4ac = 0, one repeated real root
  • If b² – 4ac < 0, no real roots (complex roots)

4. Solving Quadratic Equations

• By factorisation:
  • Find two values whose product is ac and sum is b
  • Split middle term and factorise in pairs
• By completing the square (as shown above)
• By quadratic formula:
  • x = (-b ± √(b² – 4ac)) / 2a

Example:

• x² + 4x + 3 = 0
  = (x + 1)(x + 3) = 0
  ⇒ x = -1, -3

5. Quadratic Inequalities

• Step-by-step:
  1. Solve the equation to find critical values
  2. Sketch the parabola or use sign analysis
  3. Identify required intervals based on inequality sign

Example:

• Solve x² – 4 < 0
  ⇒ x² < 4 ⇒ -2 < x < 2

6. Simultaneous Equations (One Linear, One Quadratic)

• Solve a linear and a quadratic equation together
• Method:
  1. Substitute expression from the linear equation into the quadratic
  2. Solve resulting quadratic
  3. Back-substitute to find the other variable

Example:

• y = 2x + 1
• y = x²
  ⇒ x² = 2x + 1
  ⇒ x² – 2x – 1 = 0
  ⇒ Solve for x, then find y

7. Quadratic in Disguise

• Quadratics in terms of a function of x:
  • e.g., x⁴ – 5x² + 4 = 0
    Let y = x²
    ⇒ y² – 5y + 4 = 0
    ⇒ y = 1 or 4 ⇒ x = ±1, ±2
• Also applies to trigonometric quadratics:
  • e.g., tan²x – tanx – 2 = 0
    Solve as quadratic in tanx

8. Graphs of Quadratic Functions

• Shape: Parabola
• Opens:
  • Upwards if a > 0
  • Downwards if a < 0

Key features:

• Vertex: (h, k) from completed square form
• Axis of symmetry: x = -b/2a
• y-intercept: (0, c)
• x-intercepts (roots): solve ax² + bx + c = 0

9. Example Problems

Q1. Find the vertex of y = 3x² – 6x + 2
⇒ Complete the square:
y = 3(x² – 2x) + 2 = 3(x – 1)² – 1
⇒ Vertex = (1, -1)

Q2. Solve the inequality x² – 3x + 2 > 0
⇒ x² – 3x + 2 = 0 ⇒ (x – 1)(x – 2) = 0
⇒ x < 1 or x > 2

Q3. Solve 2x + y = 3 and x² + y² = 13
⇒ y = 3 – 2x
Substitute: x² + (3 – 2x)² = 13
⇒ Solve for x, then y