Quadratic Equations — The Cricket Match
Quadratic Equations
The Cricket Match
Every ball has a trajectory. Every equation has roots. Find them before the over ends.
Standard Form
The pitch — ax² + bx + c = 0 sets the field
Standard Form
The pitch — ax² + bx + c = 0 sets the field
Roots / Zeros
The stumps — where the ball hits ground
Roots / Zeros
The stumps — where the ball hits ground
Discriminant
The pitch report — tells you what to expect
Discriminant
The pitch report — tells you what to expect
The Three Methods
Quadratic Formula
x = (-b ± √(b²-4ac)) / 2a
Works always — your six over long-on
Factorisation
ax² + bx + c = a(x-r₁)(x-r₂)
Elegant when it works — like a perfect cover drive
Completing the Square
a(x + b/2a)² + (c - b²/4a) = 0
The methodical approach — rotating strike
Key Relations
Sum of roots
r₁ + r₂ = -b/a
Partnership total
Product of roots
r₁ × r₂ = c/a
Strike rate product
Discriminant
D = b² - 4ac
The pitch report
Find the Roots Common
Solve by factorisation
1. Write in standard form 2. Find two numbers whose product = ac and sum = b 3. Split middle term and factorise
Forgetting to check if leading coefficient is 1. Always divide through first.
Solve using quadratic formula
1. Identify a, b, c 2. Calculate discriminant 3. Substitute into formula
Sign errors with -b. Write it as -(b), not -b.
Nature of Roots Common
Find k for equal roots
1. Set D = 0 2. Solve b² - 4ac = 0 for k
Not expanding (b)² properly when b contains k.
Find range of k for real roots
1. Set D ≥ 0 2. Solve the inequality
Flipping inequality sign when dividing by negative.
Word Problems Moderate
Age / Number problems
1. Let unknown = x 2. Form equation from conditions 3. Solve and reject negative/non-integer roots
Accepting both roots without checking if they make sense in context.
Area / Geometry problems
1. Express dimensions in terms of x 2. Use area/perimeter formula 3. Solve and reject negative lengths
Not rejecting the negative root for physical quantities.
Common Mistakes
1. Writing x = -b ± √(b²-4ac) / 2a without brackets → divides only √ by 2a
2. Forgetting ± gives TWO roots
3. Computing D = b² - 4ac with wrong sign on 4ac
How to Fix
1. Always write: x = (-b ± √D) / (2a) with full brackets
2. Circle the ± symbol as a reminder: solve twice
3. Write D = b² - 4(a)(c) and substitute with brackets around each value
SuPeR — Sum = -b/a, Product = c/a, Roots from quadratic formula
If you forget the formula, complete the square on ax² + bx + c = 0. The formula derives itself in 4 lines.
What does D > 0 mean?
5 cards — tap to flip
If 2x² - 8x + k = 0 has equal roots, what is k?
For equal roots, D = 0. So b² - 4ac = 0 → 64 - 8k = 0 → k = 8.