Quadratic Equations — Detailed Notes

Chapter 4 | NCERT Class 10 Mathematics

Study Diagrams (Visual Revision)

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Quadratic Equations — Decision Map

Quadratic Equations — Formula Strip

Nature of Roots from Discriminant

4.1 Introduction

Setting a quadratic polynomial equal to zero gives a quadratic equation. Example (prayer hall): breadth $x$ m, length $2 x + 1$ m, area $300$ m² $\Rightarrow$

$2 x^{2} + x - 300 = 0$

Historically, completing the square led to the quadratic formula (Sridharacharya); this chapter focuses on factorisation and nature of roots (with the formula).

4.2 Quadratic Equations

Definition

$a x^{2} + b x + c = 0, a, b, c \in R, a \neq = 0$

is the standard form. Any $p (x) = 0$ with $de g p = 2$ is quadratic; rewrite in descending powers to see standard form.

Representing situations (Example 1 style)

Translate word data into an equation, then rearrange to $a x^{2} + b x + c = 0$ .

Marbles / consecutive quantities often give product equations.
Cost $\times$ number $=$ total cost gives $x (55 - x) = 750$ type models.

Checking whether an equation is quadratic (Example 2)

Always simplify first.

Given look	After simplify	Verdict
$(x - 2)^{2} + 1 = 2 x - 3$	$x^{2} - 6 x + 8 = 0$	Quadratic
$x (x + 1) + 8 = (x + 2) (x - 2)$	$x + 12 = 0$	Not quadratic
$x (2 x + 3) = x^{2} + 1$	$x^{2} + 3 x - 1 = 0$	Quadratic
$(x + 2)^{3} = x^{3} - 4$	$x^{2} + 2 x + 2 = 0$	Quadratic (cubic terms cancel)

Better Method / Strategy

Do not judge by appearance — expand, cancel, then check degree $2$ with $a \neq = 0$ .

If leading terms cancel, the degree may drop.

4.3 Solution by Factorisation

Roots

A real number $α$ is a root if $a α^{2} + b α + c = 0$ . Roots of the equation $=$ zeroes of the polynomial. A quadratic has at most two roots.

Method

Factor $a x^{2} + b x + c$ into two linear factors; set each factor $= 0$ .

Worked idea (Example 3): $2 x^{2} - 5 x + 3 = 0$ .

Split $- 5 x$ as $- 2 x - 3 x$ (since $(- 2 x) (- 3 x) = 6 x^{2} = 2 x^{2} \cdot 3$ ):

$2 x^{2} - 2 x - 3 x + 3 = (2 x - 3) (x - 1) = 0 \Rightarrow x = \frac{3}{2} or x = 1$

Worked idea (Example 4): $6 x^{2} - x - 2 = (3 x - 2) (2 x + 1) = 0 \Rightarrow x = \frac{2}{3}$ or $x = - \frac{1}{2}$ .

Worked idea (Example 5 — repeated): $3 x^{2} - 22 x + 23 / ?$ type yields equal factors $\Rightarrow$ repeated root.

Worked idea (Example 6 — application): $2 x^{2} + x - 300 = 0 \Rightarrow (x - 12) (2 x + 25) = 0$ . Breadth $x = 12$ m (reject $- 12.5$ ); length $= 25$ m.

Better Method / Strategy — Factorisation vs Formula

Use factorisation when… Use quadratic formula when…

Integers split cleanly Messy $a c$ , or $D$ not a perfect square of an integer split

Exam says “by factorisation” Exam says “by quadratic formula” / nature of roots

Quick mental check after guessing factors Need exact surd roots

4.4 Nature of Roots

Quadratic formula

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

provided $b^{2} - 4 a c \geq 0$ (for real roots in this chapter).

Discriminant $D = b^{2} - 4 a c$

Discriminant	Nature of roots
$D > 0$	Two distinct real roots
$D = 0$	Two equal real roots (both $- b / (2 a)$ )
$D < 0$	No real roots

Worked idea (Example 7): $2 x^{2} - 4 x + 3 = 0$ : $D = 16 - 24 = - 8 < 0$ $\Rightarrow$ no real roots.

Worked idea (Example 8 — geometry): Pole on circular park diameter $13$ ; difference of distances $7$ . Pythagoras $\Rightarrow$ $x^{2} + 7 x - 60 = 0$ , $D = 289 > 0$ , $x = 5$ (reject $- 12$ ). Distances $5$ m and $12$ m from the gates.

Worked idea (Example 9): $D = 0$ $\Rightarrow$ equal roots $x = - b / (2 a)$ .

Typical $k$ -questions: For equal roots, set $D = 0$ and solve for $k$ .

Application pattern (board favourite)

Define the unknown with units.
Build a relation (area, time $=$ distance/speed, age products, Pythagoras, …).
Bring to $a x^{2} + b x + c = 0$ .
Solve by the required method.
Reject roots that break the story (negative length, speed $\leq 0$ , etc.).
State the final answer with units.

Speed/time skeleton: If speed $x$ covers distance $D$ , and speed $x - k$ takes $t$ hours more:

$\frac{D}{x - k} - \frac{D}{x} = t \Rightarrow clear denominators \Rightarrow quadratic in x$

Connecting to Chapter 2

After finding roots $α, β$ , quickly check:

$α + β = - \frac{b}{a}, α β = \frac{c}{a}$

This catches sign errors in factorisation.

Exam-Oriented Analysis

Aspect	Detail
Weight	Among the heaviest algebra chapters (often 6+ marks across paper)
Question types	Is it quadratic?; factorise; nature of roots; find $k$ for equal roots; applications
Common mistakes	Forgetting $a \neq = 0$ ; using formula when $D < 0$ without saying “no real”; keeping negative length
Scoring tips	Box final roots; show $D$ calculation; write “discard … because …”

Question-type table

Style	Typical marks	Examiner focus
Check if quadratic	1	Correct simplification
Roots by factorisation	2–3	Clear split + both roots
Nature of roots / find roots if real	2–3	$D$ with conclusion
Find $k$ for equal roots	2	$D = 0$ solved fully
Word problem	3–4	Setup + solve + reject invalid root

Common mistakes (expanded)

Declaring quadratic before expanding.
Writing formula as $\frac{- b \pm D}{a}$ (missing $2$ ).
Saying “roots are imaginary” instead of “no real roots” (Class 10 language).
Keeping $x = - 12$ as a length.
Sign error when $b$ is already negative (remember the formula starts with $- b$ ).

Scoring tips

Show $D = \dots =__$ on its own line.
After factorisation, write “therefore $x = \dots$ or $x = \dots$ ”.
In applications, add: “Since $x$ denotes …, we discard …”.

Method chooser

Equation given
  → Simplify to standard form
  → Need only nature? Compute D
  → Need roots & factors nice? Factorise
  → Need roots & D≥0 awkward? Quadratic formula
  → Application? Solve → filter context

Equal-roots mini map

$D = 0 \Rightarrow b^{2} = 4 a c \Rightarrow x = - \frac{b}{2 a} (repeated)$