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Quadratic Equations Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Q: What is a quadratic equation in Class 11 Maths?

A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. It is one of the most fundamental topics in algebra and is essential for JEE and board exams. The solutions of a quadratic equation are called its roots.

Q: What is the sum and product of roots of a quadratic equation?

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum of roots is α + β = -b/a and the product of roots is αβ = c/a. These are known as Vieta's formulas and are very useful for solving problems without finding the actual roots.

Q: What is the quadratic formula?

The quadratic formula gives the roots of ax² + bx + c = 0 as x = (-b ± √(b² - 4ac)) / 2a. It is derived by completing the square and works for all quadratic equations, including those with complex roots.

Q: Can I download the Quadratic Equations formula sheet PDF for free?

Yes. You can download the complete Quadratic Equations formula sheet PDF for free using the download button on this page. It covers standard form, discriminant, nature of roots, Vieta's formulas, graphing parabolas, and common problem types in one comprehensive place for quick revision before JEE and board exams.

By Rohit Gupta Aug 22, 2026 16 min read

Quadratic Equations Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Quadratic Equations — Competishun

Quadratic Equations Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Standard Form · Discriminant · Nature of Roots · Vieta's Formulas · Graphing Parabolas

Download Formula Sheet PDF

Quadratic Equations is one of the most fundamental and scoring chapters in Class 11 Maths. It carries significant weightage in JEE and is a guaranteed source of marks in board exams. The chapter deals with polynomial equations of degree two and their solutions, which are called roots.

This chapter covers the standard form of a quadratic equation, the quadratic formula, the discriminant and its role in determining the nature of roots, Vieta's formulas (sum and product of roots), graphing quadratic functions (parabolas), and solving word problems. Understanding these concepts is essential for success in algebra and beyond.

This page gives you the complete guide to Quadratic Equations with all concepts explained in depth. You will find clear definitions, formulas, visual diagrams, and practice questions. Download the free PDF below and keep it handy for quick revision before your JEE Main, board exams, or any other test.

Quadratic Formulax = (-b ± √D)/2a

DiscriminantD = b² - 4ac

Vieta's FormulasSum & Product of Roots

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What is a Quadratic Equation?

Definition: A quadratic equation is a polynomial equation of degree two. The general form is ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.

The word "quadratic" comes from the Latin word "quadratus," meaning square. Quadratic equations are called so because the highest power of the variable is 2 (the square).

Standard Form

ax² + bx + c = 0, where a is the coefficient of x², b is the coefficient of x, and c is the constant term. A quadratic equation always has exactly two roots (which may be real or complex, distinct or equal).

Roots/Solutions

The values of x that satisfy the equation are called the roots or solutions of the quadratic equation. They can be found using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a.

Key Insight: A quadratic equation can have at most two roots. If it has more than two roots, it becomes an identity (true for all x) rather than an equation. This is a fundamental property of polynomial equations.[reference:0][reference:1]

Glossary of Quadratic Equations Terms — Complete A to Z

Before diving deep into each topic, let's understand the key terminology used in this chapter:

Term	Definition
Quadratic Equation	A polynomial equation of degree two: ax² + bx + c = 0, a ≠ 0.
Root	A value of x that satisfies the quadratic equation. Also called a zero or solution.
Discriminant (D)	D = b² - 4ac. It determines the nature of the roots.[reference:2]
Quadratic Formula	x = (-b ± √(b² - 4ac)) / 2a. Gives the roots of the quadratic equation.[reference:3]
Vieta's Formulas	For roots α and β: α + β = -b/a and αβ = c/a.[reference:4]
Parabola	The graph of a quadratic function y = ax² + bx + c. It is a U-shaped curve.
Vertex	The turning point of the parabola. It is either the maximum or minimum point.[reference:5]
Axis of Symmetry	A vertical line through the vertex that divides the parabola into two mirror images. x = -b/(2a).[reference:6]
Real Roots	Roots that are real numbers (not involving √(-1)).
Complex Roots	Roots that are non-real complex numbers, occurring in conjugate pairs for equations with real coefficients.[reference:7]
Equal Roots	Both roots are the same (D = 0). Also called repeated or coincident roots.[reference:8]
Distinct Roots	The two roots are different from each other (D > 0).[reference:9]
Mastering these terms is essential for understanding Quadratic Equations. They will be used throughout this guide.

The Quadratic Formula — Solving Any Quadratic

Definition: The quadratic formula provides a direct method to find the roots of any quadratic equation ax² + bx + c = 0. It is derived by completing the square.

Quadratic Formula

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

The ± sign indicates that there are two roots: one with +√D and one with −√D.[reference:10]

Derivation by Completing the Square

Step 1: ax² + bx + c = 0 → 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: x² + (b/a)x + (b/2a)² = (b/2a)² - c/a
Step 4: Factor LHS: (x + b/2a)² = (b² - 4ac) / 4a²
Step 5: Take square root: x + b/2a = ±√(b² - 4ac) / 2a
Step 6: Solve for x: x = (-b ± √(b² - 4ac)) / 2a[reference:11]

Important: The quadratic formula works for every quadratic equation, including those with complex roots. It is the most reliable method when factorisation is not straightforward.

Discriminant and Nature of Roots

Definition: The discriminant of a quadratic equation ax² + bx + c = 0 is D = b² - 4ac. It determines the nature of the roots.[reference:12][reference:13]

D > 0

Two distinct real roots

The parabola intersects the x-axis at two points.[reference:14]

D = 0

Two equal real roots (repeated root)

The parabola touches the x-axis at one point.[reference:15]

D < 0

No real roots (complex conjugates)

The parabola does not intersect the x-axis.[reference:16]

Types of Roots — Detailed Table

Discriminant (D)	Nature of Roots	Explanation
D > 0	Real and Unequal (Distinct)	Two different real roots. If D is a perfect square, roots are rational; otherwise, they are irrational.[reference:17]
D = 0	Real and Equal	One repeated root (double root). The parabola touches the x-axis at the vertex.[reference:18]
D < 0	Imaginary (Complex Conjugates)	No real roots. Roots are of the form p + iq and p - iq.[reference:19]
For rational coefficients, if D > 0 and is a perfect square, the roots are rational. If D > 0 but not a perfect square, the roots are irrational and conjugate surds.[reference:20]

Complex Roots — Conjugate Pairs

When the coefficients of a quadratic equation are real and D < 0, the roots are complex conjugates. If one root is α = p + iq, the other root is β = p - iq.[reference:21]

Key Insight: The discriminant is the key to understanding the nature of roots. It tells us whether the roots are real, equal, or complex, without actually solving the equation.[reference:22]

Vieta's Formulas — Sum and Product of Roots

Definition: For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots are given by Vieta's formulas.[reference:23][reference:24]

Sum of Roots

α + β = -b/a

Negative ratio of coefficient of x to coefficient of x²[reference:25]

Product of Roots

αβ = c/a

Ratio of constant term to coefficient of x²[reference:26]

Difference of Roots

α - β = √D / a

Where D = b² - 4ac[reference:27]

Symmetric Functions of Roots

α² + β²

(α + β)² - 2αβ

Square of sum minus twice product[reference:28]

α³ + β³

(α + β)³ - 3αβ(α + β)

Sum of cubes formula[reference:29]

α² + β² + αβ

(α + β)² - αβ

Common JEE expression

Important: Vieta's formulas are extremely useful for solving problems without finding the actual roots. They are frequently tested in JEE and board exams.[reference:30]

Forming a Quadratic Equation from its Roots

Definition: If α and β are the roots of a quadratic equation, the equation can be formed as:
x² - (α + β)x + αβ = 0[reference:31][reference:32]

Example: If the roots are 2 and 3, then:
Sum = 5, Product = 6
Equation: x² - 5x + 6 = 0

Special Cases

Reciprocal roots: If the roots are α and 1/α, the equation is ax² + bx + a = 0 (c = a).
Equation with reciprocal roots: Reverse the coefficients: ax² + bx + c → cx² + bx + a[reference:33]

Graphing Quadratic Functions — The Parabola

Definition: The graph of a quadratic function y = ax² + bx + c is a U-shaped curve called a parabola. The shape and position of the parabola depend on the coefficients.[reference:34][reference:35]

Graph of y = ax² + bx + c

The parabola opens upward when a > 0 (minimum at vertex) and downward when a < 0 (maximum at vertex).[reference:36]

Key Features of a Parabola

Feature	Formula	Description
Vertex (Turning Point)	x = -b/(2a)	The point where the parabola changes direction. It is the minimum (if a > 0) or maximum (if a < 0).[reference:37][reference:38]
Axis of Symmetry	x = -b/(2a)	A vertical line that divides the parabola into two mirror images.[reference:39]
y-intercept	c	The point where the parabola crosses the y-axis (0, c).
x-intercepts (Roots)	Solutions of ax² + bx + c = 0	The points where the parabola crosses the x-axis. Determined by the discriminant.[reference:40]
The vertex is the most important point on the parabola. It gives the maximum or minimum value of the function.[reference:41]

Sign of Quadratic Expression

If a > 0 and D < 0: The expression is positive for all real x (positive definite).[reference:42]
If a < 0 and D < 0: The expression is negative for all real x (negative definite).[reference:43]
If D > 0: The expression changes sign at the roots.[reference:44]

Key Insight: The graph of a quadratic function is a parabola. The coefficient a determines the direction (opens up if a > 0, down if a < 0), and the discriminant determines the number of x-intercepts.[reference:45]

Word Problems — Applications of Quadratic Equations

Definition: Quadratic equations are used to model many real-world situations, including area problems, projectile motion, profit maximisation, and optimisation problems.[reference:46]

Common Types of Word Problems

Area Problems: Finding dimensions of rectangles, triangles, or circles given area and perimeter relationships.
Number Problems: Finding two numbers given their sum and product, or sum of squares.
Projectile Motion: Height of an object thrown upward as a function of time (h = -gt² + v₀t + h₀).
Profit/Loss Problems: Maximising profit or minimising cost.
Work Problems: Time taken by individuals or machines to complete a task.

Example: The sum of two numbers is 10 and their product is 24. Find the numbers.
Let the numbers be x and 10-x.
x(10-x) = 24 → x² - 10x + 24 = 0 → (x-4)(x-6) = 0
The numbers are 4 and 6.

Important: When solving word problems, always check the feasibility of the solutions in the context of the problem. Negative or zero solutions may be rejected if they don't make sense physically.

Common Roots and Location of Roots — JEE Advanced Topics

Definition: Two quadratic equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 are said to have a common root if there exists a value x that satisfies both equations.[reference:47]

Conditions for Common Roots

Exactly one common root: (a₁b₂ - a₂b₁)(b₁c₂ - b₂c₁) = (c₁a₂ - c₂a₁)²[reference:48][reference:49]
Both roots common: a₁/a₂ = b₁/b₂ = c₁/c₂[reference:50][reference:51]

Location of Roots

Location of roots problems ask: given f(x) = ax² + bx + c with roots α ≤ β, determine conditions so that roots satisfy specific positioning relative to real numbers k, k₁, k₂.[reference:52]

Condition	Requirements
k lies between the roots	D > 0 and a·f(k) < 0[reference:53]
Both roots > k	D ≥ 0, a·f(k) > 0, and vertex > k[reference:54]
Both roots < k	D ≥ 0, a·f(k) > 0, and vertex < k[reference:55]
Exactly one root in (k₁, k₂)	D > 0 and f(k₁)·f(k₂) < 0[reference:56]
These are among the most analytically rich JEE Main maths questions in this chapter.[reference:57]

Key Insight: Common roots and location of roots are advanced topics that are frequently tested in JEE Advanced. They require a deep understanding of the relationship between the coefficients and the roots.

Practice Questions — From JEE and Boards

Question	Answer
Q1: Find the roots of x² - 5x + 6 = 0.	(x-2)(x-3) = 0 → x = 2, 3.[reference:58]
Q2: What is the discriminant of 2x² + 3x - 2 = 0?	D = 3² - 4(2)(-2) = 9 + 16 = 25.[reference:59]
Q3: Find the sum and product of roots of x² - 3x - 4 = 0.	Sum = 3, Product = -4.[reference:60]
Q4: Solve x² + 2 = 0.	x² = -2 → x = ±√2 i.[reference:61]
Q5: For what value of k does x² + kx + 9 = 0 have equal roots?	D = 0 → k² - 36 = 0 → k = ±6.[reference:62]
Q6: Find the quadratic equation with roots 2 and -3.	Sum = -1, Product = -6 → x² + x - 6 = 0.
Q7: What is the vertex of y = x² - 4x + 3?	x = -(-4)/(2·1) = 2, y = 4 - 8 + 3 = -1 → (2, -1).
Q8: If α and β are roots of 2x² - 5x + 3 = 0, find α + β and αβ.	α + β = 5/2, αβ = 3/2.
Practise these types of questions to become comfortable with applying Quadratic Equations concepts in exam scenarios.

All Quadratic Equations Formulas at a Glance

Formula	What It Means
ax² + bx + c = 0	Standard form of quadratic equation
x = (-b ± √D) / 2a	Quadratic formula[reference:63]
D = b² - 4ac	Discriminant[reference:64]
α + β = -b/a	Sum of roots (Vieta's formula)[reference:65]
αβ = c/a	Product of roots (Vieta's formula)[reference:66]
α - β = √D / a	Difference of roots[reference:67]
x² - (α + β)x + αβ = 0	Form equation from roots[reference:68]
Vertex: x = -b/(2a)	x-coordinate of vertex[reference:69]
y_ext = (4ac - b²) / 4a	Extremum value (min/max)[reference:70]
α² + β² = (α + β)² - 2αβ	Sum of squares of roots[reference:71]
α³ + β³ = (α + β)³ - 3αβ(α + β)	Sum of cubes of roots[reference:72]
Memorise these formulas for Quadratic Equations. They are the key to scoring full marks in this chapter.

Common Mistakes in Quadratic Equations

Forgetting that a ≠ 0: If a = 0, the equation becomes linear, not quadratic.[reference:73]
Misapplying the quadratic formula: Make sure to correctly identify a, b, and c, and be careful with signs.
Forgetting the ± sign: The quadratic formula gives two roots: one with +√D and one with −√D.
Confusing the discriminant conditions: D > 0 → real and distinct; D = 0 → real and equal; D < 0 → complex conjugates.[reference:74]
Using the wrong formula for sum and product: Sum = -b/a, Product = c/a (not b/a or -c/a).[reference:75]
Not checking the nature of roots: Always check the discriminant before assuming the nature of the roots.

Golden Rule: In Quadratic Equations, always write the equation in standard form (ax² + bx + c = 0) before applying any formula. Identify a, b, and c correctly, and then use the quadratic formula or Vieta's formulas.

Why Quadratic Equations Matters for JEE and Boards

Foundation for higher mathematics: Quadratic Equations is the building block for calculus, coordinate geometry, and many other topics.
High weightage: This chapter appears in 1-2 questions in JEE Main and is a guaranteed source of marks in board exams.
Conceptual clarity: This chapter rewards students who understand the definitions and properties rather than just memorizing formulas.
Practical relevance: Quadratic equations are used everywhere, from physics to economics to engineering.

Why this guide helps: A comprehensive Quadratic Equations guide with all concepts, definitions, formulas, and practice questions saves you time during revision and helps you quickly recall everything during the exam. You won't need to look anywhere else.

Get the Complete Quadratic Equations PDF for Free

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Frequently Asked Questions — Quadratic Equations

What is a quadratic equation in Class 11 Maths?

A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. It is one of the most fundamental topics in algebra and is essential for JEE and board exams. The solutions of a quadratic equation are called its roots.[reference:76]

What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation ax² + bx + c = 0 is given by D = b² - 4ac. It determines the nature of the roots: if D > 0, roots are real and distinct; if D = 0, roots are real and equal; if D < 0, roots are imaginary (complex conjugates).[reference:77][reference:78]

What is the sum and product of roots of a quadratic equation?

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum of roots is α + β = -b/a and the product of roots is αβ = c/a. These are known as Vieta's formulas and are very useful for solving problems without finding the actual roots.[reference:79][reference:80]

What is the quadratic formula?

The quadratic formula gives the roots of ax² + bx + c = 0 as x = (-b ± √(b² - 4ac)) / 2a. It is derived by completing the square and works for all quadratic equations, including those with complex roots.[reference:81][reference:82]

Can I download the Quadratic Equations formula sheet PDF for free?

Yes. You can download the complete Quadratic Equations formula sheet PDF for free using the download button on this page. It covers standard form, discriminant, nature of roots, Vieta's formulas, graphing parabolas, and common problem types in one comprehensive place for quick revision before JEE and board exams.

Quadratic Equations Class 11: Complete Guide, All Formulas & Free PDF Download (JEE & Boards)

Download the Quadratic Equations Complete Guide PDF

What is a Quadratic Equation?

Standard Form

Roots/Solutions

Glossary of Quadratic Equations Terms — Complete A to Z

The Quadratic Formula — Solving Any Quadratic

Derivation by Completing the Square

Discriminant and Nature of Roots

Types of Roots — Detailed Table

Complex Roots — Conjugate Pairs

Vieta's Formulas — Sum and Product of Roots

Sum of Roots

Product of Roots

Difference of Roots

Symmetric Functions of Roots

α² + β²

α³ + β³

α² + β² + αβ

Forming a Quadratic Equation from its Roots

Special Cases

Graphing Quadratic Functions — The Parabola

Key Features of a Parabola

Sign of Quadratic Expression

Word Problems — Applications of Quadratic Equations

Common Types of Word Problems

Common Roots and Location of Roots — JEE Advanced Topics

Conditions for Common Roots

Location of Roots

Practice Questions — From JEE and Boards

All Quadratic Equations Formulas at a Glance

Common Mistakes in Quadratic Equations

Why Quadratic Equations Matters for JEE and Boards

Get the Complete Quadratic Equations PDF for Free