Chapter 4 of your Class 10 Maths textbook is short and single-minded: it teaches you to handle quadratic equations, anything of the form \( ax^2 + bx + c = 0 \), where \( a \neq 0 \). The chapter opens with the prayer-hall problem — a hall of 300 m² whose length is one metre more than twice its breadth — which lands on the equation \( 2x^2 + x – 300 = 0 \).
The three exercises then drill one skill each: recognising the form, solving by factorisation, and judging the nature of the roots. This page tells you which exercise holds your question and which method it wants; the worked solutions live one level down on each exercise page.
| Exercise | What you get |
|---|---|
| Exercise 4.1 | Solved |
| Exercise 4.2 | Solved |
| Exercise 4.3 | Solved |
Every Exercise in Chapter 4, and What It Tests
The chapter has three exercises and thirteen questions in all. The table below routes you to the right one by the type of question you have.
Exercise 4.1 — Recognising a Quadratic (page 4)
Two questions, twelve parts. Question 1 hands you eight equations and asks which are genuinely quadratic; the catch is that you must expand and simplify first, because an apparent cubic can collapse to a quadratic and an apparent quadratic to a linear equation.
Question 2 turns four word problems — a rectangular plot of 528 m², consecutive integers, ages, a train’s speed — into the standard form \( ax^2 + bx + c = 0 \).
Exercise 4.2 — Solving by Factorisation (page 7)
Six questions. Question 1 is the skill drill: five quadratics to solve by splitting the middle term, deliberately spanning odd coefficient forms — a surd coefficient like \( \sqrt{2}x^2 \), a fraction coefficient like \( \frac{1}{8} \), and a perfect-square trinomial where the two factors come out identical.
For the surd, treat \( \sqrt{2} \) as an ordinary number in the split; for the fraction, work step by step with \( \frac{1}{8} \); for the perfect square, expect the factor to repeat. Questions 2–6 are word problems — numbers with a given sum and product, consecutive integers, a right triangle, a cottage-industry cost problem — where you first build the equation and then factorise it.
Exercise 4.3 — Nature of Roots and the Discriminant (page 10)
Five questions, and the focus shifts from solving to judging. Question 1 finds the nature of the roots of three equations from the discriminant \( b^2 – 4ac \); Question 2 goes the other way, finding \( k \) so that a pair of roots is equal.
Questions 3–5 are “is it possible” design problems — a mango grove, two friends’ ages, a rectangular park — that live or die on the sign of the discriminant before any root is found.
The Methods and Results That Run Through the Chapter
Two techniques carry the whole chapter, and each exercise drills its own. Exercise 4.2 runs on factorisation by splitting the middle term; Exercise 4.3 runs on the quadratic formula and the discriminant.
Splitting the middle term works only when \( ax^2 + bx + c \) splits into two linear factors — you factor, then set each factor to zero. Exercise 4.2 stays purely on this method; Exercise 4.3 switches to the formula because it must handle quadratics that resist a clean split.
When the split is not obvious, the quadratic formula covers every case. The roots of \( ax^2 + bx + c = 0 \) are:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
The expression \( b^2 – 4ac \) is the discriminant, and its sign decides everything about the roots:
| Discriminant \( b^2 – 4ac \) | Nature of roots | Example equation |
|---|---|---|
| \( b^2 – 4ac \gt 0 \) | Two distinct real roots | \( x^2 – 5x + 6 = 0 \) |
| \( b^2 – 4ac = 0 \) | Two equal real roots, \( x = -\frac{b}{2a} \) | \( x^2 – 2x + 1 = 0 \) |
| \( b^2 – 4ac \lt 0 \) | No real roots | \( x^2 + x + 1 = 0 \) |
Two results are worth fixing firmly. A zero of the quadratic polynomial \( ax^2 + bx + c \) is the same number as a root of the equation \( ax^2 + bx + c = 0 \) — the link back to the quadratic polynomials of Chapter 2. And a quadratic equation has at most two roots, one per linear factor.
Common Mistakes That Cost Marks in Chapter 4
- Judging a quadratic before simplifying. An equation can look quadratic and reduce to linear, and a cubic lookalike can reduce to quadratic — the textbook’s Example 2 makes exactly this point, and Exercise 4.1 Question 1 carries both traps. Fix: expand every bracket, collect like terms, then compare with \( ax^2 + bx + c = 0 \).
- Keeping roots that cannot be real quantities. When the variable stands for a breadth, a speed, a count or a distance, a negative root is meaningless — the prayer-hall example rejects a root because a breadth cannot be negative, and the circular-park example discards a negative distance. Fix: test each root against the physical meaning before writing the final answer.
- Botching the middle-term split. In factorisation, the two parts must multiply to \( ac \) and add to \( b \). Splitting on the sum alone makes the factorisation come out wrong. Fix: check both conditions, then factor by grouping.
- Misreading the discriminant sign. A single sign slip in \( b^2 – 4ac \) when \( b \) or \( c \) is negative can flip a “two real roots” answer into “no real roots”. Fix: compute the discriminant slowly, bracketing negative terms.
- Forgetting that \( a \neq 0 \). If \( a = 0 \), the equation is linear, not quadratic, and the formula does not apply. Fix: state \( a \) first and check it is non-zero.
How to Find the Exercise You Need
Match your question to its exercise, then cross-check the textbook page numbers in the table above.
- Checking whether an equation is quadratic: expand, simplify, compare with \( ax^2 + bx + c = 0 \). That is Exercise 4.1 Question 1.
- Turning a word problem into an equation: Exercise 4.1 Question 2 — rectangular plot, consecutive integers, ages, train speed.
- Solving a quadratic: Exercise 4.2 — Question 1 drills the factorisation, Questions 2–6 solve word problems after building the equation.
- Judging the nature of roots or finding \( k \): Exercise 4.3 — read the sign of \( b^2 – 4ac \) first.
If your question is not one of these types, it may belong to the neighbouring chapter on Pair of Linear Equations in Two Variables (Chapter 3). When you finish, Arithmetic Progressions (Chapter 5) builds on the same expansion discipline.
You can verify every question against the official NCERT textbook — chapter 4 sits on pages 39 to 48 of the Class 10 Mathematics book — or move up to the full set of Class 10 Maths solutions, the Class 10 hub for all subjects, or the site index.
| Exercise | What you get |
|---|---|
| Exercise 4.1 | Solved |
| Exercise 4.2 | Solved |
| Exercise 4.3 | Solved |
Frequently Asked Questions
How many exercises are there in NCERT Class 10 Maths Chapter 4 Quadratic Equations?
Three: Exercise 4.1 (2 questions, 12 parts), Exercise 4.2 (6 questions) and Exercise 4.3 (5 questions). Exercise 4.1 checks whether an equation is quadratic, 4.2 solves by factorisation, and 4.3 judges the nature of roots through the discriminant.
Which exercise in chapter 4 has word problems I need to turn into quadratic equations?
Exercise 4.1 Question 2 asks you to form equations from four situations — a plot, consecutive integers, ages and a train’s speed. Exercise 4.2 Questions 2–6 then take word problems and solve them once you have built the equation.
When should I use factorisation instead of the quadratic formula?
When the quadratic splits neatly into linear factors, splitting the middle term is fastest (Exercise 4.2). When it does not, or you are not sure, the quadratic formula always works (Exercise 4.3) — just check that the discriminant is non-negative first.
Why does Exercise 4.3 keep asking whether a situation is possible?
Those “is it possible” questions (Q3–Q5) ask whether an equation with real roots exists at all. You answer by checking the sign of the discriminant: if \( b^2 – 4ac \lt 0 \), there are no real roots and the situation is impossible. You only hunt for actual roots after that check passes.
Reference: NCERT textbooks (CBSE).