Converting Quadratic Forms: Factored to Standard

Introduction

• The conversion involves moving from factored form (a(x - p)(x - q)) to standard form (ax^2 + bx + c).
• Key point: The 'a' value remains constant in both forms.

Example Problem 1

• Factored Form: (-3(x - 6)(x + 1))
  • 'a' value: (-3)

Steps to Convert

1. FOIL Method:

   • Multiply binomials:
     • First: (x \times x = x^2)
     • Outside: (x \times 1 = 1x)
     • Inside: (-6 \times x = -6x)
     • Last: (-6 \times 1 = -6)
   • Combine like terms:
     • Combine (1x) and (-6x) to get (-5x)
     • Expression becomes: (x^2 - 5x - 6)

2. Distribute 'a' value:

   • Multiply the trinomial by (-3):
     • (-3 \times x^2 = -3x^2)
     • (-3 \times (-5x) = 15x)
     • (-3 \times (-6) = 18)
   • Resulting in standard form: (-3x^2 + 15x + 18)

Key Insights

• From factored to standard:
  • X-intercepts can be identified from factored form as (p) and (q).
  • Y-intercept in standard form is the 'c' value.
• For the first example:
  • X-intercepts: 6, -1
  • Y-intercept (from standard form): 18

Example Problem 2

• Factored Form: (-1/2(x + 4)(x + 10))
  • 'a' value: (-1/2)

Conversion

1. FOIL Method:

   • Multiply binomials:
     • First: (x \times x = x^2)
     • Outside: (10x)
     • Inside: (4x)
     • Last: (4 \times 10 = 40)
   • Combine like terms:
     • Combine (10x) and (4x) to get (14x)
     • Expression becomes: (x^2 + 14x + 40)

2. Distribute 'a' value:

   • Multiply the trinomial by (-1/2):
     • (-1/2 \times x^2 = -1/2x^2)
     • (-1/2 \times 14x = -7x)
     • (-1/2 \times 40 = -20)
   • Resulting in standard form: (-1/2x^2 - 7x - 20)

Conclusion

• The 'a' value remains constant through conversion.
• Converting from factored to standard form involves simply applying the FOIL method, combining like terms, and distributing the 'a' value.
• Practice involves identifying x-intercepts from factored form and y-intercepts from standard form.