The expression x² + 8x + 16 can be factorised as (x + 4)² — a perfect square trinomial. This uses the identity a² + 2ab + b² = (a + b)², where a = x and b = 4. Similarly, x² − 8x + 16 = (x − 4)² using the identity a² − 2ab + b² = (a − b)².
x² + 8x + 16 = (x + 4)² — perfect square trinomial.
Identity used: a² + 2ab + b² = (a+b)², where a=x, b=4.
x² − 8x + 16 = (x − 4)² using a² − 2ab + b² = (a−b)².
Check: middle term (8x) = 2 × x × 4 = 8x ✓; last term = 4² = 16 ✓.
Alternative method: split 8x as 4x+4x, then group and factorise.
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:
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)
x² + 8x + 16 is a perfect square trinomial. Using a² + 2ab + b² = (a+b)² with a=x, b=4: x² + 8x + 16 = (x+4)². Verify: (x+4)² = x²+8x+16 ✓.
x² − 8x + 16 = (x − 4)². Using a² − 2ab + b² = (a−b)² with a=x, b=4. Verify: (x−4)² = x²−8x+16 ✓.
A trinomial ax² + bx + c is a perfect square if: the first term is a perfect square, the last term is a perfect square, and the middle term equals 2×√(first term)×√(last term).
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