In the WBBSE Class 10 Madhyamik Mathematics curriculum (Ganit Prakash / গণিত প্রকাশ), Quadratic Equations in One Variable constitutes the foundational pillar of secondary algebra. A quadratic equation in one variable $x$ is defined as any second-degree polynomial equation that can be expressed in the canonical standard form: $$ax^2 + bx + c = 0$$ where $a, b, c \in \mathbb{R}$ are real numerical coefficients with the strict constraint that $a \neq 0$. If $a = 0$, the second-degree term vanishes, degenerating into a linear equation ($bx + c = 0$). By the Fundamental Theorem of Algebra, every quadratic equation possesses exactly two roots (which may be real and distinct, real and coincident, or complex conjugates).
To obtain the roots of any arbitrary quadratic equation, ancient Indian mathematician Sreedhar Acharya (শ্রীধর আচার্য) developed the technique of completing the square, deriving the universal Quadratic Formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ The quantity under the radical, $D = b^2 - 4ac$, is known as the Discriminant (নিরূপক / विविक्तकर). The discriminant completely characterizes the nature of roots without requiring explicit solution: if $D > 0$, the roots are real and unequal (distinct); if $D = 0$, the roots are real and equal ($x = -b/2a$); if $D < 0$, no real roots exist. Furthermore, if $a, b, c$ are rational and $D$ is a positive perfect square, the roots are strictly rational.
Furthermore, if $\alpha$ and $\beta$ represent the two roots of $ax^2 + bx + c = 0$, the foundational relations connecting the roots with coefficients are: the sum of roots $\alpha + \beta = -\frac{b}{a} = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2}$, and the product of roots $\alpha\beta = \frac{c}{a} = \frac{\text{constant term}}{\text{coefficient of } x^2}$. Any quadratic equation can be reconstructed directly from its roots using: $$x^2 - (\alpha + \beta)x + \alpha\beta = 0$$ Mastering these algebraic relations empowers students to solve symmetric functions of roots, determine unknown coefficients, and formulate real-world word problems involving speed-distance-time, geometric dimensions, and financial rates.
