Factorize x² + 8x + 16

Expression: x² + 8x + 16

Recognise the pattern: a² + 2ab + b² = (a+b)²

Compare: • x² = a² → a = x • 16 = b² → b = 4 • 8x = 2ab = 2(x)(4) = 8x ✓

Since all terms match the perfect square pattern: x² + 8x + 16 = (x + 4)²

Expanded verification: (x + 4)² = x² + 2(x)(4) + 4² = x² + 8x + 16 ✓

Alternative method (splitting middle term): Find two numbers whose product = 16 and sum = 8 Those numbers: 4 and 4 (4 × 4 = 16, 4 + 4 = 8) x² + 4x + 4x + 16 = x(x + 4) + 4(x + 4) = (x + 4)(x + 4) = (x + 4)²

Expression: x² − 8x + 16

Use identity: a² − 2ab + b² = (a−b)²

Compare: • x² → a = x • 16 = b² → b = 4 • −8x = −2ab = −2(x)(4) ✓

x² − 8x + 16 = (x − 4)²

Verification: (x−4)² = x² − 8x + 16 ✓

Summary: • x² + 8x + 16 = (x + 4)² • x² − 8x + 16 = (x − 4)²

Perfect square identities:

1. a² + 2ab + b² = (a + b)²
2. a² − 2ab + b² = (a − b)²
3. a² − b² = (a + b)(a − b) [difference of squares]

More examples: • x² + 10x + 25 = (x + 5)² [b=5, 2×5=10 ✓] • x² − 6x + 9 = (x − 3)² [b=3, 2×3=6 ✓] • x² − 16 = (x+4)(x−4) [difference of squares] • 4x² + 12x + 9 = (2x + 3)² [a=2x, b=3]

How to identify a perfect square trinomial: • First term is a perfect square (x², 4x², 9x²...) • Last term is a perfect square (1, 4, 9, 16, 25...) • Middle term = 2 × √(first term) × √(last term)