Class 10 › Mathematics › Chapter

Quadratic Equations

Standard form, algebraic solutions via factorization and completing the square, Sridharacharya's quadratic formula, and discriminant analysis.

1. Quadratic Equation

A quadratic equation in the variable $x$ is an algebraic equation of degree $2$. Its general (or standard) form is:

Standard Form $$ax^2 + bx + c = 0 \qquad (a \neq 0)$$
  • $a, b,$ and $c$ are real numbers.
  • The condition $a \neq 0$ is essential; if $a = 0$, the equation reduces to the linear equation $bx + c = 0$.
  • $ax^2$ is the quadratic term, $bx$ is the linear term, and $c$ is the constant term.
2. Roots of a Quadratic Equation

A real number $\alpha$ is called a root (or zero) of the quadratic equation $ax^2 + bx + c = 0$ if substituting $x = \alpha$ satisfies the equation:

Condition for a Root $$a\alpha^2 + b\alpha + c = 0$$
Geometric Meaning: The real roots of $ax^2 + bx + c = 0$ represent the exact $x$-coordinates where the parabolic curve $y = ax^2 + bx + c$ intersects or touches the $x$-axis.
x y O x = α x = β y = ax² + bx + c
Roots $\alpha$ and $\beta$ correspond to the $x$-intercepts of the parabola.
3. Solving by Factorization (Splitting the Middle Term)

To solve $ax^2 + bx + c = 0$ by factorization:

Step-by-Step Procedure
Step 1: Compute the product of the leading coefficient and constant term: $P = a \cdot c$.
Step 2: Find two integers $p$ and $q$ such that: $$p + q = b \quad \text{and} \quad p \cdot q = a \cdot c$$
Step 3: Split the middle term $bx$ as $px + qx$: $$ax^2 + px + qx + c = 0$$
Step 4: Factor by grouping into two linear factors: $(Ax + B)(Cx + D) = 0$.
Step 5 (Zero Product Rule): Set each factor to zero to find the roots: $$Ax + B = 0 \implies x = -\frac{B}{A} \quad \text{or} \quad Cx + D = 0 \implies x = -\frac{D}{C}$$
4. Solving by Completing the Square

The method of completing the square converts a quadratic expression $ax^2 + bx + c$ into a perfect square trinomial plus a constant.

x² bx/2a bx/2a (b/2a)² x b/2a
Geometric view: adding the corner square $(b/2a)^2$ completes the square of side $(x + b/2a)$.
Derivation / Algorithmic Proof
Start with the general equation: $$ax^2 + bx + c = 0$$
Divide throughout by $a$ ($a \neq 0$): $$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$
Transpose the constant term to the right-hand side: $$x^2 + \frac{b}{a}x = -\frac{c}{a}$$
Add the square of half the coefficient of $x$, i.e., $\left(\dfrac{b}{2a}\right)^2$, to both sides: $$x^2 + 2\left(\frac{b}{2a}\right)x + \left(\frac{b}{2a}\right)^2 = \left(\frac{b}{2a}\right)^2 - \frac{c}{a}$$
Rewrite the left-hand side as a complete square: $$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} - \frac{c}{a} = \frac{b^2 - 4ac}{4a^2}$$
Taking the square root on both sides: $$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$ $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
5. Quadratic Formula / Sridharacharya's Rule

The ancient Indian mathematician Sridharacharya (c. 8th–9th century CE) derived this closed-form solution for the roots of any quadratic equation.

Quadratic Formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The expression under the radical sign determines the solvability and nature of the roots and is termed the discriminant ($D$):

The Discriminant $$D = b^2 - 4ac$$

Expressed via discriminant $D$, the two roots are:

$$\alpha = \frac{-b + \sqrt{D}}{2a}, \qquad \beta = \frac{-b - \sqrt{D}}{2a}$$
6. Number of Roots of a Quadratic Equation

By the Fundamental Theorem of Algebra, a polynomial equation of degree $n$ has exactly $n$ complex roots (counted with multiplicity). Therefore:

  • A quadratic equation ($n = 2$) has at most two distinct real roots.
  • It can have:
    • Two distinct real roots ($D > 0$).
    • Two equal real roots / one repeated root ($D = 0$).
    • No real roots / two complex conjugate roots ($D < 0$).
  • A quadratic equation cannot have more than two distinct roots. If an equation of the form $ax^2 + bx + c = 0$ is satisfied by more than two distinct values of $x$, it ceases to be an equation and becomes an identity ($a = b = c = 0$).
7. Nature of Roots with Discriminant Summary

The value of $D = b^2 - 4ac$ uniquely classifies the nature of the roots:

Classification of Roots of $ax^2 + bx + c = 0$
Value of Discriminant ($D$) Nature of Roots Roots Formula Graphical Representation ($y = ax^2 + bx + c$)
$D > 0$ Real and Distinct (Unequal) $x = \dfrac{-b \pm \sqrt{D}}{2a}$ Parabola cuts the $x$-axis at two distinct points.
$D = 0$ Real and Equal (Coincident) $x = -\dfrac{b}{2a}, \; -\dfrac{b}{2a}$ Parabola touches the $x$-axis at exactly one point.
$D < 0$ No Real Roots (Non-real / Imaginary) No real solutions Parabola lies completely above or below the $x$-axis (does not intersect).
Rational vs Irrational Roots ($a, b, c \in \mathbb{Q}$):
• If $D > 0$ and $D$ is a perfect square, the roots are rational and unequal.
• If $D > 0$ and $D$ is not a perfect square, the roots are irrational and occur in conjugate pairs: $p \pm \sqrt{q}$.