Introduction #
In algebra, quadratic equation is the equation in which the highest exponent (power) of a variable is square.
It can be written as: \[ax^2 + bx + c = 0 \quad \text{where} \quad a \ne 0\]
Here \( a, b \) and \( c \) are real numbers and \( x \) is unknown given that \( a \) is not equal to \( 0 \).
| Term | Description |
|---|---|
| \( a \) | Quadratic coefficient |
| \( b \) | Linear coefficient |
| \( c \) | Constant |
| roots | Solutions of the equation |
Standard Formula #
The standard formula to solve any given quadratic equation is $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
This yields two roots for \( x \). Check nature of roots of a quadratic equation to understand what is discriminant and how it helps us determine the nature of roots.
Graph of a Quadratic Equation #
The graph of the equation \( y = x^2 \) is a parabola. Similary the graph of a quadratic equation is also a parabola with displacements along x and/or y axes and possible stretch along y-axis depending upon the values of \( a, b \) and \( c \).
The value of \( a \) determines the shape of the curve.
| Case | Terminology | Description |
|---|---|---|
| \( a > 0 \) | Concave up | Parabola opens upward |
| \( a < 0 \) | Concave down | Parabola opens downward |
Applications of Quadratic Equation #
Quadratic equations appear in many domains of science. The equation of trajectory of a projectile motion, for example in Physics, is described as a quadratic equation:
$$ y = x \tan{\alpha} - \frac{gx^2 (1 + \tan^2{\alpha})}{2v^2} $$
where,
- \( \alpha = \) Angle of projectile above the horizontal
- \( g = \) Acceleration due to gravity
- \( v = \) Initital speed of the projectile