Polynomials & Quadratic Equations: Class 10 Complete Guide (CBSE & ICSE 2027)
Tushar Parik
Author
Two Chapters. One Algebra Core. Up to 20 Marks in Your Board Exam.
Polynomials and Quadratic Equations are the twin pillars of Class 10 Algebra in both CBSE and ICSE. Together they carry 15–20 marks — roughly one-fifth of your entire Maths paper. Whether the question asks you to find the zeroes of a polynomial, verify the relationship between zeroes and coefficients, factorise a quadratic, apply the quadratic formula, determine the nature of roots, or solve a word problem — the underlying algebra is the same. Master these two chapters and you unlock the highest-return marks on your paper. This guide covers every concept, formula, technique, and question type you need for the 2027 board exams, with step-by-step methods for each.
In This Article
- Why Polynomials & Quadratics Matter for Board Exams
- Polynomials: Degree, Types & Zeroes
- Relationship Between Zeroes and Coefficients
- Finding Zeroes: Factorisation Methods
- Quadratic Equations: Standard Form & Solutions
- The Quadratic Formula & Completing the Square
- Nature of Roots: The Discriminant
- Word Problems: Setup & Solution Strategies
- Common Board Exam Question Types
- Common Mistakes & How to Avoid Them
- Frequently Asked Questions
Why Polynomials & Quadratics Matter for Board Exams
These two chapters form the algebraic backbone of Class 10 Mathematics. Understanding why they deserve your focused attention will help you prioritise study time effectively.
- CBSE Weightage: Polynomials carries 5–7 marks and Quadratic Equations carries 8–12 marks in the CBSE Class 10 paper. Together, they often account for 15–20 marks — about 18–25% of the paper.
- ICSE Weightage: These topics appear across Section A (compulsory) and Section B (choice-based), contributing approximately 15–18 marks. ICSE favours longer, multi-step factorisation and word problems.
- High Scoring Potential: Unlike geometry proofs where presentation matters, algebra questions have definitive right answers. If your method is correct, you score full marks every time.
- Foundation for Higher Classes: Quadratic equations expand into complex numbers, polynomial functions, and calculus in Classes 11–12. A strong Class 10 foundation makes these transitions seamless.
- Competitive Exam Relevance: JEE Main, NTSE, and Olympiad papers build directly on Class 10 polynomial and quadratic concepts. Mastering them now gives you a significant head start.
Polynomials: Degree, Types & Zeroes
A polynomial is an algebraic expression made up of variables and constants, combined using addition, subtraction, and multiplication. The variable has only non-negative integer exponents. Understanding the classification of polynomials is the first step.
| Type | Degree | General Form | Example | Max Zeroes |
|---|---|---|---|---|
| Constant | 0 | c (c ≠ 0) | 7 | 0 |
| Linear | 1 | ax + b | 3x + 5 | 1 |
| Quadratic | 2 | ax² + bx + c | 2x² − 3x + 1 | 2 |
| Cubic | 3 | ax³ + bx² + cx + d | x³ − 4x + 2 | 3 |
What Is a Zero of a Polynomial?
A zero (or root) of a polynomial p(x) is any value of x for which p(x) = 0. Geometrically, zeroes are the x-coordinates where the graph of p(x) crosses or touches the x-axis. A polynomial of degree n can have at most n zeroes. For Class 10, you will work primarily with quadratic polynomials, which have at most 2 zeroes.
Geometric Meaning of Zeroes
- Linear polynomial (ax + b): The graph is a straight line that cuts the x-axis at exactly one point — the zero is x = −b/a.
- Quadratic polynomial (ax² + bx + c): The graph is a parabola. It may cross the x-axis at two distinct points (2 zeroes), touch the x-axis at one point (1 repeated zero), or not touch the x-axis at all (no real zeroes).
- Parabola direction: If a > 0, the parabola opens upward (U-shape). If a < 0, it opens downward (inverted U-shape).
Relationship Between Zeroes and Coefficients
This is one of the most important results in Class 10 Algebra. For a quadratic polynomial ax² + bx + c with zeroes α and β, the following relationships hold.
| Relationship | Formula | In Words |
|---|---|---|
| Sum of Zeroes | α + β = −b/a | Negative of coefficient of x divided by coefficient of x² |
| Product of Zeroes | αβ = c/a | Constant term divided by coefficient of x² |
Worked Example: Verify the Relationship
Consider p(x) = 2x² − 5x + 3. Here a = 2, b = −5, c = 3.
Finding zeroes: 2x² − 5x + 3 = 2x² − 2x − 3x + 3 = 2x(x − 1) − 3(x − 1) = (2x − 3)(x − 1)
Zeroes: x = 3/2 and x = 1, so α = 3/2, β = 1
Sum: α + β = 3/2 + 1 = 5/2. Also −b/a = −(−5)/2 = 5/2. ✓ Verified.
Product: αβ = (3/2)(1) = 3/2. Also c/a = 3/2. ✓ Verified.
Forming a Polynomial from Its Zeroes
If the zeroes α and β are given, you can construct the quadratic polynomial using:
p(x) = k[x² − (sum of zeroes)x + (product of zeroes)] = k[x² − (α + β)x + αβ]
where k is any non-zero constant (usually taken as 1). For example, if zeroes are 2 and −3, then p(x) = x² − (2 + (−3))x + (2)(−3) = x² + x − 6.
Useful Derived Expressions (Frequently Asked)
Board exams often ask you to find the value of expressions involving α and β without finding the actual zeroes. Use these identities:
- α² + β² = (α + β)² − 2αβ
- (α − β)² = (α + β)² − 4αβ
- 1/α + 1/β = (α + β) / αβ
- α/β + β/α = (α² + β²) / αβ = [(α + β)² − 2αβ] / αβ
- α³ + β³ = (α + β)³ − 3αβ(α + β)
Finding Zeroes: Factorisation Methods
Finding the zeroes of a quadratic polynomial ax² + bx + c means solving ax² + bx + c = 0. The three primary methods, in order of preference for board exams, are as follows.
Method 1: Splitting the Middle Term
This is the most common method in board exams. To factorise ax² + bx + c, find two numbers whose product = ac and whose sum = b. Split the middle term using these numbers, group in pairs, and extract common factors. For example, to factorise 6x² + x − 2: product = (6)(−2) = −12, sum = 1. The numbers are 4 and −3. So: 6x² + 4x − 3x − 2 = 2x(3x + 2) − 1(3x + 2) = (2x − 1)(3x + 2). Zeroes: x = 1/2 and x = −2/3.
Method 2: Common Factor Extraction
When there is no constant term (c = 0), simply take x common. For example, 3x² − 12x = 3x(x − 4). Zeroes: x = 0 and x = 4. Always check if all terms share a common factor before attempting the middle-term split — simplifying first makes the factorisation easier.
Method 3: Using Algebraic Identities
Recognise standard patterns: a² − b² = (a + b)(a − b), a² + 2ab + b² = (a + b)², and a² − 2ab + b² = (a − b)². For example, x² − 49 = (x + 7)(x − 7) and 4x² − 12x + 9 = (2x − 3)². When an expression matches an identity, factorisation is immediate — no splitting required.
Quadratic Equations: Standard Form & Solutions
A quadratic equation is any equation of the form ax² + bx + c = 0 where a ≠ 0. While polynomials and quadratic equations are closely related, the distinction matters for board exams: polynomial questions focus on zeroes and their relationships, while quadratic equation questions focus on solving and applying.
Three Methods to Solve ax² + bx + c = 0
- Factorisation: Split the middle term (covered above). Preferred when the equation factorises cleanly. Board exams often specify “solve by factorisation” explicitly.
- Completing the Square: Rewrite ax² + bx + c = 0 as a(x + p)² = q. Useful for understanding the derivation of the quadratic formula, and sometimes asked as a standalone method.
- Quadratic Formula: x = [−b ± √(b² − 4ac)] / 2a. Works for every quadratic equation, including those that do not factorise into rational factors. The most powerful general method.
The Quadratic Formula & Completing the Square
The quadratic formula is derived from the completing-the-square method and is the most universally applicable tool for solving quadratic equations.
Derivation by Completing the Square
Start with ax² + bx + c = 0 (a ≠ 0)
Step 1: Divide by a: x² + (b/a)x + c/a = 0
Step 2: Move constant: x² + (b/a)x = −c/a
Step 3: Add (b/2a)² to both sides: [x + (b/2a)]² = (b² − 4ac) / 4a²
Step 4: Take square root: x + b/2a = ±√(b² − 4ac) / 2a
Step 5: Solve for x: x = [−b ± √(b² − 4ac)] / 2a
Worked Example: Using the Quadratic Formula
Solve 2x² + 7x − 15 = 0 using the quadratic formula.
Here a = 2, b = 7, c = −15.
Discriminant D = b² − 4ac = 49 − 4(2)(−15) = 49 + 120 = 169
√D = √169 = 13
x = (−7 ± 13) / (2 × 2) = (−7 ± 13) / 4
x = (−7 + 13)/4 = 6/4 = 3/2 or x = (−7 − 13)/4 = −20/4 = −5
Nature of Roots: The Discriminant
The discriminant D = b² − 4ac determines the nature of the roots of a quadratic equation without actually solving it. This is one of the most frequently tested concepts in board exams.
| Discriminant (D) | Nature of Roots | Graphical Meaning |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola crosses the x-axis at two points |
| D = 0 | Two equal (repeated) real roots | Parabola touches the x-axis at one point |
| D < 0 | No real roots (imaginary roots) | Parabola does not intersect the x-axis |
| D is a perfect square | Two distinct rational roots | Equation factorises cleanly |
Worked Example: Discriminant Application
Find the value of k for which 2x² + kx + 3 = 0 has equal roots.
For equal roots, D = 0.
D = k² − 4(2)(3) = k² − 24 = 0
k² = 24
k = ±2√6
Key Discriminant Questions You Must Practise
- Find the value(s) of k for which the equation has equal roots (set D = 0).
- Find the value(s) of k for which the equation has real and distinct roots (set D > 0).
- Find the value(s) of k for which the equation has no real roots (set D < 0).
- Show that a given equation has real roots for all values of the parameter (prove D ≥ 0).
- Determine the nature of roots without solving the equation (calculate D and interpret).
Word Problems: Setup & Solution Strategies
Word problems are the highest-mark questions in quadratic equations — typically 4–5 marks — and the area where most students struggle. The key is a systematic approach to converting words into algebra.
Step 1: Assign Variables
Read the problem twice. Identify the unknown quantity and let it be x. Express all other related quantities in terms of x. For example, if two consecutive numbers are involved, let them be x and x + 1. If one number is 5 more than another, let them be x and x + 5.
Step 2: Form the Equation
Use the given condition (product, sum, difference, area, speed-distance-time) to set up an equation. This equation should be quadratic (ax² + bx + c = 0). If your equation is not quadratic, re-read the problem — you likely missed a relationship.
Step 3: Solve and Validate
Solve the quadratic equation by factorisation or the formula. Check both roots against the problem context: reject negative values for age, length, or count; reject fractional values for whole numbers. Always state: “Since x represents [quantity], x = [negative value] is not valid. Therefore, x = [positive value].”
Common Word Problem Categories
- Number Problems: “Find two consecutive positive integers whose product is 306.” Let the integers be x and x + 1. Then x(x + 1) = 306, giving x² + x − 306 = 0.
- Age Problems: “A man's age is the square of his son's age. Six years hence, his age will be three times his son's age.” Let son's age be x. Father's age = x². After 6 years: x² + 6 = 3(x + 6).
- Area/Geometry Problems: “The area of a rectangular plot is 528 m². Its length is one more than twice its breadth. Find the dimensions.” Let breadth = x. Length = 2x + 1. Then x(2x + 1) = 528.
- Speed-Distance-Time: “A train covers 480 km at a uniform speed. If the speed had been 8 km/h less, it would have taken 3 hours more. Find the speed.” Let speed = x. Time = 480/x. New time: 480/(x − 8) = 480/x + 3.
- Work Problems: “Two pipes fill a tank in 12 hours. The larger pipe fills 10 hours faster than the smaller pipe alone.” Let smaller pipe take x hours. Larger pipe: x − 10. Then 1/x + 1/(x − 10) = 1/12.
Common Board Exam Question Types
Board exams follow predictable patterns. Here are the question types you will encounter across both CBSE and ICSE, ordered by frequency.
- Type 1 — Find the Zeroes and Verify (3–4 marks): Given a quadratic polynomial, find its zeroes by factorisation and verify the sum and product relationships.
- Type 2 — Find the Polynomial from Zeroes (2–3 marks): Given the sum and product of zeroes (or the zeroes themselves), construct the quadratic polynomial.
- Type 3 — Solve by Factorisation (3–4 marks): Solve a quadratic equation using the splitting-the-middle-term method. Both CBSE and ICSE specify this method explicitly.
- Type 4 — Solve Using the Quadratic Formula (3–4 marks): Solve using x = [−b ± √(b² − 4ac)] / 2a. Often paired with irrational roots or “give answer correct to 2 decimal places.”
- Type 5 — Nature of Roots (2–3 marks): Find discriminant, determine nature, or find the value of k for equal/real/no-real roots.
- Type 6 — Word Problems (4–5 marks): Convert a real-world scenario into a quadratic equation, solve, and interpret. Always carries the most marks.
- Type 7 — Expressions Involving Zeroes (3 marks, ICSE favourite): Given a polynomial, find values like α² + β², 1/α + 1/β, α/β + β/α using sum-product relationships without finding the actual zeroes.
Common Mistakes & How to Avoid Them
These errors cost students the most marks. Being aware of each one dramatically reduces your error rate.
- Sign errors in the sum formula: The sum of zeroes is −b/a, NOT b/a. The negative sign is the most commonly forgotten detail. Circle it in your rough work.
- Forgetting a ≠ 0: In ax² + bx + c = 0, the coefficient ‘a’ must not be zero. If a = 0, it becomes a linear equation, not quadratic.
- Not rearranging to standard form: Before applying the formula or factorisation, always rearrange the equation to ax² + bx + c = 0. Equations like x² = 3x + 10 must become x² − 3x − 10 = 0 first.
- Accepting invalid roots in word problems: If the question asks for age, length, or speed, negative roots must be rejected with a written justification. Boards deduct marks for missing this statement.
- Confusing zeroes with coefficients: Zeroes are the values of x that make p(x) = 0. Coefficients are the numbers multiplying the variable terms. These are related (via the sum-product formulas) but not the same.
- Skipping the verification step: When the question says “find the zeroes and verify the relationship,” you must show both the sum verification and the product verification separately. Missing either costs marks.
- Discriminant arithmetic errors: In D = b² − 4ac, compute b² and 4ac separately before subtracting. Pay special attention when b or c is negative — (−5)² = 25, not −25, and 4(2)(−3) = −24, so D = 25 − (−24) = 49.
Frequently Asked Questions
Q: What is the difference between a polynomial and a quadratic equation?
A polynomial is an algebraic expression like 2x² + 3x − 5. A quadratic equation is a polynomial set equal to zero: 2x² + 3x − 5 = 0. The polynomial chapter focuses on zeroes, graphs, and the relationship between zeroes and coefficients. The quadratic equations chapter focuses on solving equations and applying them to real-world problems. Both are tested separately in board exams.
Q: How many marks do polynomials and quadratic equations carry in the CBSE/ICSE Class 10 board exam?
In CBSE, polynomials carry about 5–7 marks and quadratic equations carry 8–12 marks (total 15–20 marks out of 80). In ICSE, these topics together contribute approximately 15–18 marks. Both boards test factorisation, the quadratic formula, discriminant-based questions, and word problems every single year.
Q: When should I use factorisation versus the quadratic formula?
Use factorisation when the question explicitly asks for it or when the equation has clean integer or simple fractional roots. Use the quadratic formula when the equation does not factorise easily, when the question asks for roots “correct to 2 decimal places,” or when the discriminant is not a perfect square. If the question does not specify a method, either approach earns full marks.
Q: What does it mean when a quadratic equation has “equal roots”?
Equal roots (also called repeated roots or coincident roots) means both roots are the same. This happens when the discriminant D = b² − 4ac = 0. The single root is x = −b/2a. Graphically, the parabola just touches the x-axis at one point without crossing it. Board exams frequently ask you to find the value of a parameter k that makes the discriminant zero.
Q: How do I handle word problems where both roots seem valid?
In some word problems, both roots may be mathematically valid but only one makes sense contextually. For instance, if you are finding the speed of a train and get x = 40 or x = −12, reject the negative value. However, in some number problems, both positive and negative roots are valid. Always check: Does this value make physical/real-world sense? State your reasoning when rejecting a root to earn full marks.
Q: Can I find α² + β² without actually finding α and β?
Yes, and board exams expect this approach. Use the identity α² + β² = (α + β)² − 2αβ. Simply substitute the sum (−b/a) and product (c/a) into this formula. Similarly, 1/α + 1/β = (α + β)/αβ = (−b/a)/(c/a) = −b/c. These shortcuts save time and reduce calculation errors significantly.
Q: Are these topics important for JEE and competitive exams?
Extremely important. JEE Main and Advanced test quadratic equations extensively — including the theory of equations, location of roots, and quadratic inequalities. All of these build directly on the Class 10 foundation of discriminant analysis, sum-product relationships, and factorisation techniques. NTSE and Olympiad papers also draw heavily from these topics.
Q: How much time should I dedicate to preparing polynomials and quadratics for boards?
Plan 10–14 days of focused preparation. Days 1–3: master polynomial types, zeroes, and the sum-product relationships. Days 4–7: practise factorisation (20+ equations), the quadratic formula (10+ equations), and discriminant problems. Days 8–12: solve 15–20 word problems covering all five categories (numbers, age, area, speed, work). Days 13–14: attempt 3–4 previous year papers under timed conditions. Students who follow this plan consistently score 90%+ in these chapters.
Polynomials & Quadratics: Your Algebra Score Multiplier
These two chapters reward systematic preparation more than any other topic in Class 10 Maths. Once you master the three factorisation methods, the quadratic formula, the discriminant conditions, and the five word-problem categories, you have 15–20 marks secured before you even see the question paper. The formulas are few, the patterns are predictable, and the marks are generous. Make this chapter your strongest.
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