Quadratic Equations and Theory of Equations

Location of Roots and Sign Analysis

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Location of Roots and Sign Analysis

Location of Roots and Sign Analysis

This lesson covers the following key concepts:

  • Conditions for roots to lie in given interval
  • Conditions for a number to lie between roots
  • Conditions for both roots greater/less than k
  • Conditions for exactly one root in interval
  • Graphical interpretation of root location
  • Sign of quadratic expression f(x) = ax² + bx + c
  • Maximum and minimum values of quadratic expression

Important Formulas

  • Both roots >k:D≥0,f(k)>0,−b2a>k\text{Both roots } > k: D \geq 0, f(k) > 0, -\frac{b}{2a} > k
  • Both roots <k:D≥0,f(k)>0,−b2a<k\text{Both roots } < k: D \geq 0, f(k) > 0, -\frac{b}{2a} < k
  • k lies between roots: f(k)<0k \text{ lies between roots: } f(k) < 0
  • Exactly one root in (k1,k2):f(k1)⋅f(k2)<0\text{Exactly one root in } (k_1, k_2): f(k_1) \cdot f(k_2) < 0
  • Max/Min at x=−b2a, value =4ac−b24a\text{Max/Min at } x = -\frac{b}{2a}, \text{ value } = \frac{4ac - b^2}{4a}
  • Sign of f(x): same as a outside roots, opposite inside\text{Sign of } f(x): \text{ same as } a \text{ outside roots, opposite inside}