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 $0$.

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.

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