🎯 Learning Objectives
- Rewrite x² + bx + c in completed-square form.
- Identify the turning point from the completed square.
- Use the method to prepare for solving quadratics.
Introduction
This lesson develops Completing the Square from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
🤔 Start with an Idea
A square such as (x + 3)² expands to x² + 6x + 9. Completing the square asks: what square is hiding inside this quadratic?
📖 Key Notes
For x² + bx, take half of b and square it. Add and subtract that value so the expression stays unchanged.
x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7
🔍 Reveal the Hidden Square
x² + 8x + 3
What number belongs with x inside the bracket?
🧠 Let's Think Together
Rewrite: x² − 10x + 7.
- Half of −10 is −5.
- (x − 5)² creates x² − 10x + 25.
- Compensate: subtract 25, then add 7.
x² − 10x + 7 = (x − 5)² − 18
✍️ Worked Examples
x² + 4x + 1 = (x + 2)² − 3
x² − 6x + 11 = (x − 3)² + 2
🎯 Practice Questions
Easy
Complete the square: x² + 2x + 5.
Show answer
(x + 1)² + 4
Challenge
Complete the square: x² − 12x − 1.
Show answer
(x − 6)² − 37
⚠️ Common Mistakes
- Using b instead of half of b.
- Adding a square without subtracting it again.
- Losing the sign when b is negative.
📝 Exam Focus
When asked for a turning point, completed-square form (x − h)² + k gives the point (h, k). Notice the sign reversal inside the bracket.
Summary
- Half the x-coefficient.
- Square it.
- Add and subtract it.
- Write the first three terms as a perfect square.