Factoring ③ Formula 2 (Perfect-Square Trinomials)
Using the expansion formula (a±b)² = a² ± 2ab + b² in reverse, we can factor perfect-square trinomials.
Formula for perfect-square trinomials
a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
📘 Example ①
Factor x² + 6x + 9.
Answer:x² + 6x + 9 = x² + 2×x×3 + 3² = (x + 3)²
Factor x² + 6x + 9.
Answer:x² + 6x + 9 = x² + 2×x×3 + 3² = (x + 3)²
📘 Example ②
Factor x² − 10x + 25.
Answer:x² − 10x + 25 = x² − 2×x×5 + 5² = (x − 5)²
Factor x² − 10x + 25.
Answer:x² − 10x + 25 = x² − 2×x×5 + 5² = (x − 5)²
💡 Key points
- Check whether the first and last terms are in square form
- Check whether the middle term has the form 2ab
- Whether the expression is (a+b)² or (a−b)² is determined by the sign of the middle term
Practice Problems
- Factor x² + 8x + 16.
- Factor 4x² − 4x + 1.
Answers and Explanations
- Answer:x² + 8x + 16 = (x + 4)²
- Answer:4x² − 4x + 1 = (2x)² − 2×2x×1 + 1² = (2x − 1)²