Quadratic Formula Notes

The Quadratic Formula

• Quadratic Formula: Used to find the solutions of quadratic equations of the form $ax^2 + bx + c = 0$. The formula is:
  $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example Problem 1: Solve the equation $x^2 - 4x + 1 = 0$

  • Coefficients: $a = 1, b = -4, c = 1$

  • Calculation of the discriminant: $D = b^2 - 4ac = (-4)^2 - 4(1)(1) = 16 - 4 = 12$

  • Solutions:

  • $x = \frac{4 \pm \sqrt{12}}{2}$

  • The simplified solutions become: $x = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}$

• Example Problem 2: Solving $2x^2 + 8x - 10 = 0$

  • Coefficients: $a = 2, b = 8, c = -10$

  • Calculation of the discriminant: $D = b^2 - 4ac = 8^2 - 4(2)(-10) = 64 + 80 = 144$

  • Solutions:

  • $x = \frac{-8 \pm \sqrt{144}}{2(2)}$

  • After simplification, the solutions are: $x = \frac{-8 \pm 12}{4}$

    • This gives: $x = 1 ext{ and } x = -5$

• Another Example: Solving $x^2 - 12x + 36 = 0$

  • Coefficients: $a = 1, b = -12, c = 36$

  • Calculation of the discriminant: $D = (-12)^2 - 4(1)(36) = 144 - 144 = 0$ (indicating a double root)

  • Solutions:

  • $x = \frac{-(-12) \pm \sqrt{0}}{2(1)} = \frac{12}{2} = 6$

• Complex Solutions Example: Solve $x^2 + x - 6 = -4$

  • Rearranged to $x^2 + x + 2 = 0$

  • Coefficients: $a = 1, b = 1, c = 2$

  • Discriminant: $D = 1^2 - 4(1)(2) = 1 - 8 = -7$

  • Solutions:

  • $x = \frac{-1 \pm \sqrt{-7}}{2} = \frac{-1 \pm i\sqrt{7}}{2}$

• Final Example Problem: Solve $3x^2 + 12x + 6 = 0$

  • Coefficients: $a = 3, b = 12, c = 6$

  • Discriminant: $D = 12^2 - 4(3)(6) = 144 - 72 = 72$

  • Solutions:

  • $x = \frac{-12 \pm \sqrt{72}}{2(3)} = \frac{-12 \pm 6\sqrt{2}}{6}$

  • Thus, simplifying yields: $x = -2 \pm \sqrt{2}$