Lecture Notes: Quadratic Equations

Review of Previous Session

• Quadratic Equations Introduction
  • Started with equations of the form: (ax^2 + c = 0)
  • Reduced to (x^2 = k) leading to (x = \pm \sqrt{k}) (if (k) is positive)

Today's Focus: General Quadratic Equation

• General Form: (ax^2 + bx + c = 0)
• Methods to Solve:
  1. Factoring
  2. Quadratic Formula

Factoring Method

• Applicable when the polynomial can be factored
• Example Problem:
  • Given: (30x^2 + 7x - 2 = 0)
  • Find two numbers that multiply to (-60) and add to (7): (12) and (-5)
  • Rewrite: (30x^2 + 12x - 5x - 2 = 0)
  • Factor: ((6x)(5x + 2) - 1(5x + 2) = 0)
  • Solutions: (x = -\frac{2}{5}, \frac{1}{6})

Quadratic Formula

• Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})
• Two solutions unless (b^2 - 4ac = 0) (discriminant conditions)
• Example Problem:
  • Non-factorable: (x^2 + 4x - 7 = 0)
  • Identify: (a = 1, b = 4, c = -7)
  • Use Formula: (x = \frac{-4 \pm \sqrt{16 + 28}}{2})
  • Solutions: (x = -2 \pm \sqrt{11})

Proof of the Quadratic Formula

• Completing the Square:
  • Start with (ax^2 + bx + c = 0)
  • Factor out (a): (a(x^2 + \frac{b}{a}x) = -c)
  • Add and subtract (\frac{b^2}{4a^2}) to complete the square
  • Results in: (a(x + \frac{b}{2a})^2 - \frac{b^2}{4a} = -c)
  • Final form: ((x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2})
  • Solve for (x): (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})

Conclusion

• The quadratic formula results from completing the square.
• Next session: Another example and application of quadratic equations.

Note: Always check the discriminant (b^2 - 4ac) to determine the nature of the roots (real and distinct, real and repeated, or complex).