Lecture Notes on Solving Quadratic Equations

Key Steps in Solving Quadratic Equations

1. Set the Quadratic to Zero

   • Ensure the quadratic equation is set to zero before proceeding.

2. Identify Coefficients

   • Label the coefficients:
     • a (coefficient of x²)
     • b (coefficient of x)
     • c (constant term)
   • Example: For the equation (x^2 - 5x - 9 = 0),
     • (a = 1)
     • (b = -5)
     • (c = -9)

Determining the Discriminant

• Formula: (b^2 - 4ac)
• Calculation:
  • Substitute the values of a, b, and c into the formula.
  • Example with given values: ((-5)^2 - 4 \times 1 \times (-9) = 25 + 36 = 61)
• Interpretation of the Discriminant:
  • Since 61 is not a perfect square, the equation has two real irrational solutions.

Solving Quadratics Using the Quadratic Formula

• Quadratic Formula: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
• Steps:
  1. Calculate (b^2 - 4ac) (already found as 61).
  2. Calculate the opposite of (b):
     • If (b = -5), then (-b = 5).
  3. Substitute into the formula:
     • (x = \frac{5 \pm \sqrt{61}}{2})
• Simplifying and Expressing Solutions:
  • Solutions can be expressed as:
    • (x = \frac{5 + \sqrt{61}}{2})
    • (x = \frac{5 - \sqrt{61}}{2})
  • Ensure you understand that these are acceptable forms of the solution.

Additional Notes

• Solution Set Representation:
  • Can also write solutions as a set: ( {\frac{5 + \sqrt{61}}{2}, \frac{5 - \sqrt{61}}{2}} )
• Simplification:
  • While you could divide 2 into both terms, in this example, it cannot be simplified further.
  • If simplifiable, divide appropriately.

Conclusion

• For problems requiring solutions, use the quadratic formula.
• Recognize different forms of valid solutions.
• Practice memorizing the quadratic formula for ease of use in solving equations.

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Tip: Practice frequently to become comfortable with identifying coefficients, using the discriminant, and applying the quadratic formula efficiently.