Factoring Perfect Square Trinomials

Introduction

• Focus on factoring perfect square trinomials.

Example 1: x² + 6x + 9

• To factor, identify two numbers that multiply to 9 and add to 6.
  • Numbers: 3 and 3
  • Factorization: (x + 3)(x + 3) or (x + 3)².
• Verification:
  • x² + 3x + 3x + 9 = x² + 6x + 9.

Identifying Perfect Square Trinomials

• Check:
  • Square root of first coefficient: 1 (x²)
  • Square root of constant term: 3 (9)
• Multiply: 1 * 3 = 3, which is half of 6. Thus, it’s a perfect square trinomial.*

General Formula for Perfect Square Trinomials

• General form: a² + 2ab + b² = (a + b)²

Example 2: x² + 10x + 25

• Square root of 25 is 5.
• Check: 1 * 5 = 5, which is half of 10. Perfect square trinomial.
• Factorization: (x + 5)².*

Example 3: 4x² + 12x + 9

• Square root of 4 is 2 and square root of 9 is 3.
• Check: 2 * 3 = 6, which is half of 12. Perfect square trinomial.
• Factorization: (2x + 3)².*

Different Techniques for Non-1 Leading Coefficients

• Example: 4x² + 12x + 9
  • Multiply leading and last coefficients: 4 * 9 = 36.
  • Find numbers that multiply to 36 and add to 12: 6 & 6.
  • Rewrite: 4x² + 6x + 6x + 9.
  • Group and factor:
    • Group 1: 2x(2x + 3)
    • Group 2: 3(2x + 3)
  • Final Factorization: (2x + 3)².*

Example 4: 9a² + 30ab + 25b²

• Square roots: 3 and 5.
• Check: 3 * 5 = 15, half of 30. Perfect square trinomial.
• Factorization: (3a + 5b)².*

Confirming Factorization with Grouping

• Multiply coefficients: 9 * 25 = 225.
• Numbers for 225 that add to 30: 15 & 15.
• Replace middle term: 15ab + 15ab.
• Group and factor:
  • Group 1: 3a(3a + 5b)
  • Group 2: 5b(3a + 5b)
• Final confirmation: (3a + 5b)².*