Higher Tier • 2.2.DGrade 7-9 Separator
THE VERTEX FINDER: Completing the Square
Master the technique that reveals the hidden minimum/maximum of any quadratic
📐 THE ALGORITHM
✓ Step 1: Factor out coefficient of x² (if needed)
✓ Step 2: Half the middle term → Square it → Subtract it
✓ Step 3: Result format: a(x + b)² + c
The Vertex: The minimum/maximum is at (-b, c)
⚠️ EXAMINER TRAP ALERT
❌ WRONG: Forgetting to factor out a when a ≠ 1
✓ CORRECT: Always write a[x² + (b/a)x] + c first
📊 FREQUENCY: Very Common • 📉 MARK IMPACT: 3-4 marks
THE VISUAL TOY
2x6x12x
x²
The Big Square
3x
Rectangle 1
3x
Rectangle 2
🔲 THE MISSING CORNER
(3)² = 9
To complete the square, we added 9
So we must subtract 9
✓ COMPLETED FORM
(x + 3)² - 9
THE LOGIC TRACK
🎯 WHAT WE'RE DOING
We're trying to rearrange x² + 6x into a perfect square.
🔲 THE GEOMETRIC INSIGHT
Think of x² + 6x as building a square from pieces:
• One big x² square
• Two rectangles, each with area 3x
To complete the square, we need the missing corner:(3)² = 9
⚖️ THE BALANCE RULE
We added 9 to complete the square, but that changes the value!
Solution: Subtract 9 to keep everything balanced.
x² + 6x = (x + 3)² - 9
✓ WHY THIS MATTERS
The lowest value a square can be is 0.
So (x + 3)² is minimized when the bracket equals 0.
→ This happens when x = -3
→ The minimum value is -9
THE DYNAMIC SOLVER
COMPLETE THE SQUARE FOR:
x² +
x +
COMPLETED SQUARE FORM
2(x + 2)² + 2
VERTEX
(-2, 2)
🎓 THE COMPLETE METHOD
1️⃣ THE SETUP
• If a ≠ 1, factor it out
• Focus on x² + bx inside
2️⃣ THE MAGIC
• Half the coefficient of x
• Square that number
• Add it, then subtract it
3️⃣ THE PAYOFF
• Read vertex from form
• Check sign of 'a'
• State min/max clearly
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