Section 2.3: Quadratic Formula & Roots
This section teaches solving quadratics using the quadratic formula and interpreting roots, including real, repeated, and complex solutions.
Example 1: Solve using Quadratic Formula
Solve \( x^2 - 4x - 5 = 0 \) using the quadratic formula.
Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Here, a=1, b=-4, c=-5
Discriminant: \( (-4)^2 - 4(1)(-5) = 16 + 20 = 36 \)
Roots: \( x = \frac{4 \pm 6}{2} \Rightarrow x_1 = 5, x_2 = -1 \)
Example 2: Complex Roots
Solve \( x^2 + 4x + 8 = 0 \).
a=1, b=4, c=8 → Discriminant: \( 16 - 32 = -16 \)
Roots: \( x = \frac{-4 \pm \sqrt{-16}}{2} = \frac{-4 \pm 4i}{2} = -2 \pm 2i \)
Practice Problems
- Solve \( x^2 + 3x - 10 = 0 \) using quadratic formula
- Solve \( 2x^2 - 4x + 1 = 0 \)
- Solve \( x^2 + 6x + 10 = 0 \) and identify complex roots
- Verify roots of \( x^2 - 7x + 12 = 0 \)
- Find the nature of roots for \( x^2 + 2x + 5 = 0 \)