Solving Quadratic Equations by Factoring

Key Techniques

Difference of Perfect Squares

• Example: (x^2 - 49 = 0)
• Factor as ((x + 7)(x - 7)).
• Set each factor equal to 0:
  • (x + 7 = 0 \rightarrow x = -7)
  • (x - 7 = 0 \rightarrow x = 7)

Factoring with Greatest Common Factor (GCF)

• Example: (3x^2 - 75 = 0)
• Find GCF: 3.
• Factor: (3(x^2 - 25)).
• Use difference of squares on (x^2 - 25):
  • ((x + 5)(x - 5))
• Solve:
  • (x + 5 = 0 \rightarrow x = -5)
  • (x - 5 = 0 \rightarrow x = 5)

Quadratics with Different Leading Coefficients

• Example: (9x^2 - 64 = 0)
• Use difference of squares:
  • ((3x + 8)(3x - 8))
• Solve:
  • (3x + 8 = 0 \rightarrow x = -\frac{8}{3})
  • (3x - 8 = 0 \rightarrow x = \frac{8}{3})

Factoring Trinomials

• Example: (x^2 - 2x - 15)

• Find two numbers that multiply to -15 and add to -2:

  • Numbers are -5 and 3.

• Factor: ((x - 5)(x + 3))

• Solve:

  • (x - 5 = 0 \rightarrow x = 5)
  • (x + 3 = 0 \rightarrow x = -3)

• Example: (x^2 + 3x - 28)

• Numbers that multiply to -28 and add to 3:

  • Numbers are 7 and -4.

• Factor: ((x - 4)(x + 7))

• Solve:

  • (x - 4 = 0 \rightarrow x = 4)
  • (x + 7 = 0 \rightarrow x = -7)

Factoring Trinomials with Leading Coefficient not 1

• Example: (8x^2 + 2x - 15)
• Multiply leading coefficient by constant term: 8 * -15 = -120.
• Numbers that multiply to -120 and add to 2:
  • Numbers are 12 and -10.
• Replace middle term (2x) with (12x - 10x).
• Factor by grouping:
  • (4x(2x+3) - 5(2x+3))
  • Result: ((2x+3)(4x-5))
• Solve:
  • (2x + 3 = 0 \rightarrow x = -\frac{3}{2})
  • (4x - 5 = 0 \rightarrow x = \frac{5}{4})*

Using the Quadratic Formula

• Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})

• Example: Solve (x^2 - 2x - 15)

  • (a = 1, b = -2, c = -15)
  • Substitute into formula:
    • (x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \times 1 \times (-15)}}{2 \times 1})
    • Simplifies to: (x = \frac{2 \pm 8}{2})
  • Solutions: (x = 5) and (x = -3)

• Example: (8x^2 + 2x - 15)

  • (a = 8, b = 2, c = -15)
  • Substitute into formula:
    • (x = \frac{-2 \pm \sqrt{2^2 - 4 \times 8 \times (-15)}}{2 \times 8})
    • Simplifies to: (x = \frac{-2 \pm 22}{16})
  • Solutions: (x = \frac{5}{4}) and (x = -\frac{3}{2})

Conclusion

• Quadratic equations can be solved by factoring or using the quadratic formula.
• Practice problems with different techniques to become proficient.