Factoring Difference of Two Squares

Formula

• Expression: ( a^2 - b^2 )
• Factored Form: ( (a + b)(a - b) )
• Steps:
  1. Find the square root of ( a^2 ) (which is ( a )).
  2. Find the square root of ( b^2 ) (which is ( b )).
  3. Place them in parentheses with a plus and a minus sign.

Examples

• Example 1: ( x^2 - 36 )

  • ( x^2 \rightarrow x )
  • ( 36 \rightarrow 6 )
  • Factored form: ( (x + 6)(x - 6) )

• Example 2: ( x^2 - 49 )

  • ( x^2 \rightarrow x )
  • ( 49 \rightarrow 7 )
  • Factored form: ( (x + 7)(x - 7) )

• Example 3: ( y^2 - 64 )

  • ( y^2 \rightarrow y )
  • ( 64 \rightarrow 8 )
  • Factored form: ( (y + 8)(y - 8) )

Harder Examples

1. ( 4x^2 - 25 )

   • Square root of ( 4x^2 ) is ( 2x )
   • Square root of ( 25 ) is ( 5 )
   • Factored form: ( (2x + 5)(2x - 5) )

2. ( 9y^2 - 100 )

   • Square root of ( 9y^2 ) is ( 3y )
   • Square root of ( 100 ) is ( 10 )
   • Factored form: ( (3y + 10)(3y - 10) )

3. ( 36x^2 - 121 )

   • Square root of ( 36x^2 ) is ( 6x )
   • Square root of ( 121 ) is ( 11 )
   • Factored form: ( (6x + 11)(6x - 11) )

4. ( 49x^2 - 169y^2 )

   • Square root of ( 49x^2 ) is ( 7x )
   • Square root of ( 169y^2 ) is ( 13y )
   • Factored form: ( (7x + 13y)(7x - 13y) )

Dealing with Non-Perfect Squares

• Steps:

  1. Identify the GCF (Greatest Common Factor).
  2. Factor out the GCF.
  3. Apply the difference of squares method to the remaining expression.

• Example: ( 3x^2 - 27 )

  • GCF is 3.
  • Reduced form: ( x^2 - 9 )
  • Factored form: ( (x + 3)(x - 3) )

• Example: ( 5y^2 - 80 )

  • GCF is 5.
  • Reduced form: ( y^2 - 16 )
  • Factored form: ( (y + 4)(y - 4) )

Complex Examples

1. ( x^4 - 81 )

   • Square root of ( x^4 ) is ( x^2 )
   • Square root of ( 81 ) is ( 9 )
   • Factored form: ( (x^2 + 9)(x^2 - 9) )
   • Further factor ( x^2 - 9 ): ( (x + 3)(x - 3) )

2. ( 16x^4 - 81y^4 )

   • Initial factors: ( (4x^2 + 9y^2)(4x^2 - 9y^2) )
   • Further factor ( 4x^2 - 9y^2 ): ( (2x + 3y)(2x - 3y) )

Additional Problems

• Fractional Coefficients: Apply the same square root method adjusting for fractions.
• Using Brackets: When dealing with expressions like ((3x + 5y)^2 - (4x + 7y)^2), simplify the expression inside the brackets before applying the formula.

Key Points

• Always check for a GCF first when the square roots do not result in integers.
• Practice with various forms and coefficients to be prepared for different types of problems.

This summary provides a comprehensive understanding of factoring the difference of two squares, useful for exams and further studies.