Quadratic Equations — Cheat Sheet

Class X · Mathematics Ch.4 · 8 Panels · Quick Reference

📐 Standard Form & Basics
ax² + bx + c = 0   (a ≠ 0)
a, b, c ∈ ℝ  |  degree = 2  |  max 2 roots
QuadraticFrom Latin "quadratus" (square) — refers to the x² term
Root / ZeroValue of x satisfying the equation. Also = x-intercept of parabola y=ax²+bx+c
To convert to standard form: Expand all brackets, move everything to LHS, set RHS = 0, simplify.
✂️ Factorisation Method
Steps (Splitting the Middle Term)
  1. Compute product = a × c and note sum = b
  2. Find two numbers with that product and sum
  3. Split the middle term bx using those two numbers
  4. Group into two pairs and factor each
  5. Take out common bracket → (px+q)(rx+s) = 0
  6. Apply zero-product rule: each factor = 0
Special Cases
Difference of squares: x² − k² = (x−k)(x+k)
Perfect square: x² ± 2kx + k² = (x±k)²
Common factor: ax² + bx = x(ax+b) → one root = 0
Limitation: Works well only when D = b²−4ac is a perfect square (rational roots).
🔲 Completing the Square
Steps
  1. Divide by a:   x² + (b/a)x + c/a = 0
  2. Move constant:   x² + (b/a)x = −c/a
  3. Add (b/2a)² to both sides
  4. LHS becomes:   (x + b/2a)² = (b²−4ac)/4a²
  5. Take square root:   x + b/2a = ±√(b²−4ac)/2a
  6. Solve for x
Key identity: x² + px + (p/2)² = (x + p/2)²
Add (half of coefficient of x)² to complete the square.
This method derives the quadratic formula! Apply it to the general form ax²+bx+c=0.
🧮 Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
Shreedharacharya Formula — works for ALL quadratics
How to Apply
  1. Identify a, b, c from standard form
  2. Compute D = b² − 4ac
  3. If D ≥ 0: x = (−b + √D)/2a and x = (−b − √D)/2a
  4. If D < 0: No real roots exist
Always works — unlike factorisation, gives exact roots even when irrational.
🎯 Discriminant & Nature of Roots
D = b² − 4ac
DiscriminantNature of RootsGraph
D > 0 (perfect sq)Two distinct rational rootsCrosses x-axis at 2 points
D > 0 (not perf sq)Two distinct irrational rootsCrosses x-axis at 2 points
D = 0Two equal (repeated) rootsTouches x-axis at 1 point
D < 0No real roots (complex)Doesn't touch x-axis
"Find k for equal roots": Set D = 0 → b²−4ac = 0 → solve for k. This is a VERY common exam question.
🔗 Vieta's Formulas & Relationships
For roots α, β of ax² + bx + c = 0
Sum of roots: α + β = −b/a
Product of roots: α × β = c/a
Difference: |α − β| = √D / |a|
Forming Equations from Roots
If roots are α and β:
x² − (α+β)x + αβ = 0
i.e., x² − (sum)x + (product) = 0
Verify rootsCheck: sum of your answers = −b/a and product = c/a
Common Q"Form equation whose roots are 2α, 2β" — use new sum/product
🌍 Application Problem Types
TypeSetup PatternKey Equation
Numbersx, (S−x) or x, x+dProduct or sum of squares given
Area/Geometryx, (x+k)l × b = A or Pythagoras
Speed-TimeD/x − D/(x+k) = tCross-multiply → quadratic
Agex, (x±k)Product of ages at different times
Work1/x + 1/(x±k) = 1/TCross-multiply → quadratic
Consecutivex, x+1 or x, x+2Sum of squares, product relations
Always reject negative roots when they don't make physical sense (speed, age, length can't be negative).
Strategy: Define variable → Form equation → Solve → Reject invalid root → State answer in context.
🧠 Memory Tricks & Exam Tips
Formula Memory Aid
"Negative boy couldn't decide (±) whether to go to the radical party. But the boy was too square to miss out on 4 awesome chicks. And it was all over by 2am."
→ x = (−b ± √(b²−4ac)) / 2a
Method Selection
SituationBest Method
Simple integer roots expectedFactorisation
Leading coefficient ≠ 1, messy numbersQuadratic Formula
Asked to "derive" or show workingCompleting the Square
Only nature of roots askedJust compute D
Common Errors
Sign of c: In 2x²−3x = 5, rewrite as 2x²−3x−5=0. Here c = −5, NOT +5.
Dividing by x: Never divide both sides by x (you'll lose the root x=0). Factor instead: x(2x+3)=0.
Exam tip: Always verify roots in the ORIGINAL equation. For 5-mark questions, verification earns 1 mark.