Chapter 22: Properties of Quadratic Equations
Once we can solve a quadratic equation, we can begin to study the nature of its roots without actually solving the equation. This is a powerful tool for predicting the behavior of mathematical models.
22.1 The Discriminant
In the general quadratic equation ax² + bx + c = 0, the expression under the radical in the quadratic formula (b² - 4ac) is called the Discriminant. It tells us the "nature" of the roots.
- If b² - 4ac > 0: The roots are real and unequal.
- If b² - 4ac = 0: The roots are real and equal (a perfect square).
- If b² - 4ac < 0: The roots are imaginary.
Example: Determine the nature of the roots for 2x² - 4x + 5 = 0.
Identify coefficients: a=2, b=-4, c=5.
Calculate b² - 4ac: (-4)² - 4(2)(5) = 16 - 40 = -24.
Result: Since the discriminant is negative, the roots are imaginary.
22.2 Relations Between Roots and Coefficients
There are two beautiful relationships that exist in every quadratic equation ax² + bx + c = 0. If we let r¹ and r² be the roots:
1. The Sum of the Roots: r¹ + r² = -b/a
2. The Product of the Roots: r¹ × r² = c/a
22.3 Forming an Equation from Given Roots
If you know the roots you want, you can work backward to build the equation. If the roots are 3 and -5, the equation is found by multiplying the factors:
(x - 3)(x + 5) = 0
x² + 2x - 15 = 0
Doctor's Pro-Tip: Using the "Sum and Product" rules is a fantastic way to quickly check your work after solving a difficult quadratic. If your answers don't add up to -b/a, you know there's a sign error somewhere!