Algebra - Quadratic Equations in one variable

Standard Form: A quadratic equation in one variable $x$ is represented as $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$. Visually, the graph of a quadratic equation is a parabola. If $a > 0$, the parabola opens upwards (like a U-shape); if $a < 0$, it opens downwards (like an inverted U).

Roots of the Equation: The values of $x$ that satisfy the equation are called its roots or zeros. Geometrically, these roots represent the x-intercepts of the parabola, where the graph crosses or touches the horizontal x-axis.

Nature of Roots (The Discriminant): The value $D = b^2 - 4ac$ is called the discriminant. It determines the nature of the roots without solving the equation. Visually: if $D > 0$, the parabola cuts the x-axis at two distinct points (real and unequal roots); if $D = 0$, the vertex of the parabola touches the x-axis at exactly one point (real and equal roots); if $D < 0$, the parabola stays entirely above or below the x-axis and never touches it (imaginary roots).

Solving by Factorization: This method involves rewriting the quadratic expression $ax^2 + bx + c$ as a product of two linear factors $(px + q)(rx + s) = 0$. We usually split the middle term $bx$ into two terms such that their sum is $b$ and their product is $ac$. Once factored, we apply the Zero Product Rule: if $A \times B = 0$, then $A = 0$ or $B = 0$.

Quadratic Formula (Sridharacharya's Rule): For equations where factorization is difficult, the roots can be calculated using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula provides a direct algebraic path to find both values of $x$.

Relation between Roots and Coefficients: If $\alpha$ and $\beta$ are the two roots of $ax^2 + bx + c = 0$, then the sum of the roots is given by $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is given by $\alpha\beta = \frac{c}{a}$. These relationships allow us to form a quadratic equation if the roots are known using the structure $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$.

Equations Reducible to Quadratic Form: Some equations are not quadratic initially but can be converted into one through substitution. For example, an equation like $x^4 - 5x^2 + 4 = 0$ can be treated as a quadratic by letting $x^2 = y$, resulting in $y^2 - 5y + 4 = 0$.