VG2
kalkulus
Quiz
Completing the square
Completing the square is an elegant technique for solving quadratic equations and rewriting quadratic expressions.
Completing the square is a technique where we write a quadratic expression in the form (x + p)² + q. This makes it easier to find the vertex and solve equations.
ax² + bx + c = a(x + b/2a)² + (c - b²/4a)
Method
Step by step:
1. Factor out coefficient a from x²
2. Add and subtract (b/2a)²
3. Recognise the perfect square (x + b/2a)²
4. Simplify the constant term
1. Factor out coefficient a from x²
2. Add and subtract (b/2a)²
3. Recognise the perfect square (x + b/2a)²
4. Simplify the constant term
Examples
Example 1: Write x² + 6x + 5 as a completed square
x² + 6x + 5
= x² + 6x + 9 - 9 + 5
= (x + 3)² - 4
= x² + 6x + 9 - 9 + 5
= (x + 3)² - 4
Vertex: (-3, -4)
Example 2: Solve x² + 4x - 12 = 0 by completing the square
x² + 4x - 12 = 0
(x + 2)² - 4 - 12 = 0
(x + 2)² = 16
x + 2 = ±4
x = 2 or x = -6
(x + 2)² - 4 - 12 = 0
(x + 2)² = 16
x + 2 = ±4
x = 2 or x = -6
Advantage: Completing the square gives you the vertex (p, q) directly without needing the vertex formula separately!
Algebra is generous; she often gives more than is asked of her.
— Jean le Rond d'Alembert (1717–1783)
🧠 Test yourself
Question 1 of 5
Write x² + 8x + 7 as a completed square
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