Solving Quadratic Equations by Factoring

Techniques for Factoring Quadratic Equations

1. Difference of Perfect Squares: For equations of the form x^2 - a^2 = 0.

   • Example: x^2 - 49 = 0
     • Square root of x^2 is x, and 49 is 7.
     • Factor as: (x + 7)(x - 7).
     • Solve: Set each factor to zero: x + 7 = 0 and x - 7 = 0.
     • Solutions: x = -7, x = 7.

2. Factoring by Taking Out GCF: Factor out the greatest common factor (GCF) first.

   • Example: 3x^2 - 75 = 0
     • GCF is 3: 3(x^2 - 25) = 0.
     • Now use difference of squares: (x + 5)(x - 5) = 0.
     • Solutions: x = -5, x = 5.

3. Factoring Trinomials with Leading Coefficient = 1: Find two numbers that multiply to ac and add to b.

   • Example: x^2 - 2x - 15
     • ac = -15, b = -2.
     • Numbers: -5 and 3.
     • Factor as: (x - 5)(x + 3).
     • Solutions: x = 5, x = -3.

4. Factoring When Leading Coefficient is Not 1:

   • Example: 8x^2 + 2x - 15
     • Multiply a and c: 8 * (-15) = -120.
     • Numbers that multiply to -120 and add to 2: 12, -10.
     • Replace 2x with 12x and -10x, then factor by grouping.
     • Factor: 4x(2x + 3) - 5(2x + 3) = (2x + 3)(4x - 5).
     • Solutions: Solve each factor: x = -3/2, x = 5/4.*

Solving Using the Quadratic Formula

• Formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a

• Example: Solve x^2 - 2x - 15

  • a = 1, b = -2, c = -15
  • Substitute into formula: x = [2 ± sqrt(4 + 60)] / 2
  • Solution: x = 5 and x = -3

• Example: Solve 8x^2 + 2x - 15

  • a = 8, b = 2, c = -15
  • Substitute into formula: x = [-2 ± sqrt(4 + 480)] / 16
  • Solution: x = 5/4 and x = -3/2

Summary

• Quadratic equations can be solved by factoring or using the quadratic formula.
• Both methods often lead to the same solutions, but the approach used depends on the form of the equation.
• Practice identifying the most efficient method for different forms of quadratic equations.