CBSE · Class 6-12

Class 10 · Mathematics (Basic)

Unit 4 · Class 10Mathematics (Basic)

Quadratic Equations

Standard form, factorisation, completing the square, quadratic formula, nature of roots and word problems.

Lessons

6 concepts

1
Standard Form and Roots
A quadratic equation is of the form $ax^2 + bx + c = 0$ with $a \ne 0$ . A number $\alpha$ is a root (or zero) if $a\alpha^2 + b\alpha + c = 0$ . Every quadratic has at most two roots. Real coefficients need not yield real…
2
Solving by Factorisation
Rewrite $ax^2 + bx + c$ as a product of two linear factors: split the middle term $b$ as $m + n$ with $mn = ac$ . Then factor by grouping. E.g. $6x^2 + 11x + 3$ : $18 = 9\cdot 2$ , $9+2 = 11$ ; factors $(2x+3)(3x+1)$ .
3
Completing the Square
Divide by $a$ : $x^2 + (b/a)x + c/a = 0$ . Add and subtract $(b/2a)^2$ : $\left(x + \dfrac{b}{2a}\right)^2 = \dfrac{b^2 - 4ac}{4a^2}$ . Taking square roots gives the quadratic formula.
4
Nature of Roots via Discriminant
$D = b^2 - 4ac$ . If $D > 0$ the roots are real and distinct; if $D = 0$ they are real and equal ( $x = -b/2a$ ); if $D < 0$ they are complex conjugates (no real roots). The graph of $y = ax^2 + bx + c$ cuts, touches, or mi…
5
Sum and Product of Roots
If $\alpha, \beta$ are the roots of $ax^2 + bx + c = 0$ : $\alpha + \beta = -b/a$ , $\alpha\beta = c/a$ . This helps in constructing quadratics or answering questions like 'find $\alpha^2 + \beta^2$ ', which equals $(\alpha …
6
Word-Problem Strategies
Common contexts: area/perimeter, speed–time, age, work–rate, consecutive numbers. Set a variable, translate the wording into a quadratic, solve, and reject any answer that is negative or otherwise infeasible in the physi…

Skills to practise

Skill · medium

Quadratic Equations — Practice