Definition of a Quadratic Equation
A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term where the variable is squared, but no terms where the variable is raised to a higher power. It sets this polynomial equal to zero.
Standard Form and Components
The standard form of a quadratic equation is `ax² + bx + c = 0`, where 'x' represents an unknown variable, and 'a', 'b', and 'c' are known numerical coefficients, with 'a' not equal to zero. The `ax²` is the quadratic term, `bx` is the linear term, and `c` is the constant term.
A Practical Example
Consider the equation `x² - 5x + 6 = 0`. Here, `a=1`, `b=-5`, and `c=6`. The solutions (or roots) for 'x' in this specific equation are `x=2` and `x=3`, as substituting these values back into the equation makes it true.
Importance and Applications
Quadratic equations are fundamental in many fields. They describe the path of projectile motion in physics, model parabolic shapes in engineering designs (like satellite dishes or bridge arches), and are used in economics to calculate maximum profit or minimum cost.