Quadratic Equations Class 11 for IIT JEE

Q: What are Vieta’s formulas for a quadratic equation?

If α and β are the roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. They let you find expressions such as α² + β², α³ + β³ and 1/α + 1/β without finding α and β, which is how most JEE problems on Quadratic Equations Class 11 are solved.

Q: How do you find the nature of roots using the discriminant?

Calculate D = b² − 4ac. If D > 0, the roots are real and distinct. If D = 0, they are real and equal. If D < 0, they are complex conjugates. When the coefficients are rational and D is a perfect square, the roots are rational.

Q: How do you form a quadratic equation from given roots?

Use x² − (sum of roots)x + (product of roots) = 0. For roots 2 + √3 and 2 − √3, the sum is 4 and the product is 1, so the equation is x² − 4x + 1 = 0. If one root is irrational or complex and the coefficients are rational or real, the other root is its conjugate.

Q: How do you find the maximum or minimum value of a quadratic expression?

For f(x) = ax² + bx + c, the extreme value is −D/4a at x = −b/2a. It is a minimum if a > 0 and a maximum if a < 0. For example, x² − 6x + 11 has a minimum value of 2 at x = 3.

Q: How do you solve a quadratic inequality?

In Quadratic Equations Class 11 notes, find the roots and use the sign of a. For a > 0, f(x) > 0 outside the roots and f(x) < 0 between them. For example, x² − 5x + 6 < 0 gives 2 < x < 3.

Q: Is quadratic equations in the NEET syllabus?

No. NEET covers Physics, Chemistry and Biology only, so quadratic equations are needed for CBSE boards and JEE (and other exams with Mathematics), and also for solving equilibrium and kinematics problems in Chemistry and Physics.

Q: Where can I find more Class 11 Maths notes on eSaral?

eSaral has handwritten notes, JEE Main Maths revision notes, NCERT books and chapter-wise PYQs, linked with these Quadratic Equations Class 11 notes on this page.

Team eSaral

eSaral Academic and Editorial Team

19th October 2026

9 min read

Summary

Quadratic Equations Class 11 notes cover the standard form, the quadratic formula and discriminant, nature and sign of roots, Vieta’s formulas, forming equations from given roots, root transformations, the graph and extreme values of a quadratic expression, quadratic inequalities, location of roots and common roots. This page gives tables, solved examples and handwritten notes for CBSE boards and JEE Main.

Quadratic Expression Important Questions - IIT JEE - eSaral

In NCERT Class 11 Mathematics, quadratic equations are part of the chapter Complex Numbers and Quadratic Equations. It is Chapter 4 in the rationalised book (Chapter 5 in older books). Boards focus on solving equations, including those with complex roots, while JEE Main also asks about the nature and location of roots and the relations between roots and coefficients. NEET does not test Mathematics. These Quadratic Equations Class 11 notes cover both levels.

What Is a Quadratic Equation? Standard Form in Quadratic Equations Class 11 Notes

A quadratic equation is a polynomial equation of degree 2. Its standard form is:

ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0

If a = 0, the equation becomes linear. A quadratic equation always has exactly two roots (counting repeated roots), and these may be real or complex.

Equation	a, b, c	Quadratic?
x² − 5x + 6 = 0	1, −5, 6	Yes
2x² + 3x − 2 = 0	2, 3, −2	Yes
x² − 4 = 0	1, 0, −4	Yes
x³ − 2x + 1 = 0	Degree 3	No (cubic)
0·x² + 2x − 1 = 0	a = 0	No (linear)

Solving a Quadratic Equation: Three Methods in Quadratic Equations Class 11 Notes

Factorisation: x² − 5x + 6 = (x − 2)(x − 3) = 0, so x = 2 or x = 3.
Completing the square: write the equation as (x + b/2a)² = D/4a².
Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. It works for every quadratic equation, whether the roots are rational, irrational or complex.

Quadratic Equations Class 11

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Discriminant and Nature of Roots in Quadratic Equations Class 11 Notes

The expression D = b² − 4ac is the discriminant. It decides the nature of the roots without solving the equation.

Value of D	Nature of roots
D > 0	Real and distinct
D = 0	Real and equal (x = −b/2a)
D < 0	Complex conjugates: (−b ± i√\|D\|)/2a
D > 0 and a perfect square (a, b, c rational)	Rational and distinct
D > 0 and not a perfect square (a, b, c rational)	Irrational conjugates, such as p ± √q

If a, b and c are real, complex roots come in conjugate pairs. If a, b and c are rational, irrational roots of the form p + √q come in conjugate pairs. This is why one root decides the other when you form an equation.

Sign of the Roots Without Solving

Condition	Conclusion
D ≥ 0, α + β > 0, αβ > 0	Both roots positive
D ≥ 0, α + β < 0, αβ > 0	Both roots negative
αβ < 0	Roots of opposite signs
αβ = 0 and α + β ≠ 0	One root is zero

Vieta’s Formulas: Roots and Coefficients in Quadratic Equations Class 11 Notes

If α and β are the roots of ax² + bx + c = 0, then:

Sum of roots: α + β = −b/a
Product of roots: αβ = c/a

Expression	In terms of (α + β) and αβ
α² + β²	(α + β)² − 2αβ
α³ + β³	(α + β)³ − 3αβ(α + β)
(α − β)²	(α + β)² − 4αβ
\|α − β\|	√D / \|a\|
1/α + 1/β	(α + β)/αβ = −b/c
α²β + αβ²	αβ(α + β) = −bc/a²

Tip from the eSaral Maths team: In JEE problems, you rarely need the roots themselves. Express the required quantity in terms of α + β and αβ, and substitute −b/a and c/a. This is faster and avoids arithmetic errors.

Forming a Quadratic Equation from Given Roots in Quadratic Equations Class 11 Notes

If the roots are α and β, the equation is x² − (α + β)x + αβ = 0.

Roots 2 + √3 and 2 − √3: sum = 4 and product = 4 − 3 = 1, so the equation is x² − 4x + 1 = 0.
Root 3 + 2i (real coefficients): the other root is 3 − 2i. Sum = 6 and product = 13, so the equation is x² − 6x + 13 = 0.

Transformations of Roots in Quadratic Equations Class 11 Notes

If α and β are the roots of ax² + bx + c = 0, find the equation of the transformed roots by a substitution in x.

Required roots	Substitute	New equation
−α, −β	x → −x	ax² − bx + c = 0
1/α, 1/β	x → 1/x	cx² + bx + a = 0
α + k, β + k	x → x − k	a(x − k)² + b(x − k) + c = 0
kα, kβ	x → x/k	ax² + bkx + ck² = 0
α², β²	x → √x	a²x² + (2ac − b²)x + c² = 0

Graph and Extreme Values of a Quadratic Expression in Quadratic Equations Class 11 Notes

The graph of f(x) = ax² + bx + c is a parabola with its vertex at x = −b/2a.

If a > 0, the parabola opens upwards, and the minimum value is −D/4a at x = −b/2a.
If a < 0, the parabola opens downwards, and the maximum value is −D/4a at x = −b/2a.
If D < 0 and a > 0, then f(x) > 0 for every real x. If D < 0 and a < 0, then f(x) < 0 for every real x.

Quadratic Inequalities in Quadratic Equations Class 11 Notes

For a > 0 and real roots α < β:

f(x) > 0 when x < α or x > β (outside the roots)
f(x) < 0 when α < x < β (between the roots)

Example: x² − 5x + 6 < 0 means (x − 2)(x − 3) < 0, so 2 < x < 3.

Location of Roots in Quadratic Equations Class 11 Notes

For f(x) = ax² + bx + c with a > 0 and real roots, the conditions are:

Required condition	Conditions to apply
Both roots greater than k	D ≥ 0, f(k) > 0, and −b/2a > k
Both roots less than k	D ≥ 0, f(k) > 0, and −b/2a < k
k lies between the roots	f(k) < 0 (in general, a·f(k) < 0)
Both roots in (p, q)	D ≥ 0, f(p) > 0, f(q) > 0, and p < −b/2a < q
Exactly one root in (p, q)	f(p)·f(q) < 0

Common Roots of Two Quadratic Equations in Quadratic Equations Class 11 Notes

For a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0:

Both roots common: a₁/a₂ = b₁/b₂ = c₁/c₂.
Exactly one common root: (a₁c₂ − a₂c₁)² = (a₁b₂ − a₂b₁)(b₁c₂ − b₂c₁).

Solved Examples from Quadratic Equations Class 11 Notes

Example 1 (equal roots): Find k for which x² + kx + 9 = 0 has equal roots.

D = k² − 36 = 0, so k = ±6.

Example 2 (Vieta’s formulas): If α and β are the roots of 2x² − 5x + 1 = 0, find α² + β².

α + β = 5/2 and αβ = 1/2. α² + β² = (5/2)² − 2(1/2) = 25/4 − 1 = 21/4.

Example 3 (transformation): If α and β are the roots of x² − 5x + 3 = 0, form the equation with roots 1/α and 1/β.

Replace x by 1/x: cx² + bx + a = 0 gives 3x² − 5x + 1 = 0.

Example 4 (both roots positive): Find the values of k for which x² − 4x + k = 0 has two real positive roots.

D ≥ 0 gives 16 − 4k ≥ 0, so k ≤ 4. The sum 4 > 0, and the product k > 0. So 0 < k ≤ 4.

Example 5 (common root): Find a if x² + x + a = 0 and x² + ax + 1 = 0 have a common root.

Subtracting the equations gives (1 − a)(x − 1) = 0. So either a = 1 (the equations are identical) or x = 1, which gives 1 + 1 + a = 0, so a = −2. Therefore a = 1 or a = −2.

Example 6 (minimum value): Find the minimum value of x² − 6x + 11.

D = 36 − 44 = −8, so the minimum value is −D/4a = 8/4 = 2, at x = 3.

Common Mistakes in Quadratic Equations Class 11 Problems

Mistake	Correct approach
Forgetting that a ≠ 0	Check that the coefficient of x² is not zero, especially in parameter problems
Using D > 0 when the question allows equal roots	Check whether it says “real”, which means D ≥ 0
Using only D to place roots	Also check f(k), the sum or the vertex, as in the location of roots table
Writing the wrong sign in the sum of roots	α + β = −b/a, with the minus sign
Applying the conjugate-root rule with non-real coefficients	The rule needs real (or rational) coefficients
Forgetting to check the common root in both equations	Substitute the value that you find into both equations

How to Study Quadratic Equations Class 11 Notes

Identify D, α + β and αβ in every problem before you solve it, and then write the condition in terms of these three.
Solve the NCERT examples and exercises of Complex Numbers and Quadratic Equations. Download the book from eSaral’s NCERT books for Class 11 page.
Practise sub-topic wise: nature of roots, Vieta’s formulas, location of roots and common roots.
Revise with eSaral’s JEE Main Maths revision notes, and then solve eSaral’s JEE Main chapter-wise PYQs.

Frequently Asked Questions

What is the quadratic formula for Class 11?

For ax² + bx + c = 0 (a ≠ 0), the roots are x = (−b ± √(b² − 4ac))/2a. The expression D = b² − 4ac is the discriminant. If D > 0, the roots are real and distinct. If D = 0, they are real and equal. If D < 0, they are complex conjugates. eSaral’s Quadratic Equations Class 11 notes give examples of each.

What are Vieta’s formulas for a quadratic equation?

If α and β are the roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. They let you find expressions such as α² + β², α³ + β³ and 1/α + 1/β without finding α and β, which is how most JEE problems on Quadratic Equations Class 11 are solved.

How do you find the nature of roots using the discriminant?

Calculate D = b² − 4ac. If D > 0, the roots are real and distinct. If D = 0, they are real and equal. If D < 0, they are complex conjugates. When the coefficients are rational and D is a perfect square, the roots are rational.

How do you form a quadratic equation from given roots?

Use x² − (sum of roots)x + (product of roots) = 0. For roots 2 + √3 and 2 − √3, the sum is 4 and the product is 1, so the equation is x² − 4x + 1 = 0. If one root is irrational or complex and the coefficients are rational or real, the other root is its conjugate.

How do you find the maximum or minimum value of a quadratic expression?

For f(x) = ax² + bx + c, the extreme value is −D/4a at x = −b/2a. It is a minimum if a > 0 and a maximum if a < 0. For example, x² − 6x + 11 has a minimum value of 2 at x = 3.

How do you solve a quadratic inequality?

In Quadratic Equations Class 11 notes, find the roots and use the sign of a. For a > 0, f(x) > 0 outside the roots and f(x) < 0 between them. For example, x² − 5x + 6 < 0 gives 2 < x < 3.

Is quadratic equations in the NEET syllabus?

No. NEET covers Physics, Chemistry and Biology only, so quadratic equations are needed for CBSE boards and JEE (and other exams with Mathematics), and also for solving equilibrium and kinematics problems in Chemistry and Physics.

Where can I find more Class 11 Maths notes on eSaral?

eSaral has handwritten notes, JEE Main Maths revision notes, NCERT books and chapter-wise PYQs, linked with these Quadratic Equations Class 11 notes on this page.

Team eSaral

eSaral Academic and Editorial Team

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Quadratic Equations Class 11 for IIT JEE

Summary

What Is a Quadratic Equation? Standard Form in Quadratic Equations Class 11 Notes

Solving a Quadratic Equation: Three Methods in Quadratic Equations Class 11 Notes

Quadratic Equations Class 11

Discriminant and Nature of Roots in Quadratic Equations Class 11 Notes

Sign of the Roots Without Solving

Vieta’s Formulas: Roots and Coefficients in Quadratic Equations Class 11 Notes

Forming a Quadratic Equation from Given Roots in Quadratic Equations Class 11 Notes

Transformations of Roots in Quadratic Equations Class 11 Notes

Graph and Extreme Values of a Quadratic Expression in Quadratic Equations Class 11 Notes

Quadratic Inequalities in Quadratic Equations Class 11 Notes

Location of Roots in Quadratic Equations Class 11 Notes

Common Roots of Two Quadratic Equations in Quadratic Equations Class 11 Notes

Solved Examples from Quadratic Equations Class 11 Notes

Common Mistakes in Quadratic Equations Class 11 Problems

How to Study Quadratic Equations Class 11 Notes

Frequently Asked Questions

Team eSaral

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