Introduction to Inequalities
Quadratic Equations
A quadratic polynomial is an expression of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are real constants. Generally, we assume \( a \neq 0 \), as \( a = 0 \) would simplify the expression to \( bx + c \), which is easier to study.
Proposition 1: Sign of the Polynomial
Let \( a, b, c \in \mathbb{R} \) with \( a \neq 0 \). Consider the quadratic polynomial \( P(x) = ax^2 + bx + c \).
- If \( b^2 - 4ac < 0 \), then \( P(x) \) has the sign of \( a \) for all \( x \in \mathbb{R} \) and never equals zero.
- If \( b^2 - 4ac = 0 \), then \( P(x) \) has the sign of \( a \) for all \( x \in \mathbb{R} \) and equals zero at \( x = \frac{-b}{2a} \).
- If \( b^2 - 4ac > 0 \), then \( P(x) \) has two roots \( x_1 \) and \( x_2 \), and its sign is the same as \( a \) for \( x \in (-\infty, x_1) \cup (x_2, \infty) \), and opposite for \( x \in [x_1, x_2] \).
Proposition 2: Roots of the Polynomial
The roots \( x_1 \) and \( x_2 \) of the quadratic polynomial \( P(x) \) are given by the quadratic formula:
\[ x_1 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}, \quad x_2 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \]
Proposition 3: Interpretation of the Discriminant
Based on the value of the discriminant \( \Delta = b^2 - 4ac \):
- If \( \Delta < 0 \), the polynomial has no real roots.
- If \( \Delta = 0 \), the polynomial has exactly one real root, meaning the parabola touches the x-axis.
- If \( \Delta > 0 \), the polynomial has two distinct real roots, meaning the parabola crosses the x-axis at two points.