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CBSE Class 10 Maths Chapter 4 Notes: Quadratic Equations

Complete local notes for CBSE Class 10 Maths Quadratic Equations: standard form, factorisation method, completing square, quadratic formula, discriminant, word problems, examples, practice questions, and exam tips.

Updated 23 May 20267 min read

Chapter Overview

Quadratic Equations is one of the most scoring Algebra chapters in Class 10 Maths. It is important because the same idea appears later in graphs, coordinate geometry, applications, and higher classes.

This chapter teaches:

  • What a quadratic equation is
  • How to solve by factorisation
  • How to solve by completing the square
  • How to use the quadratic formula
  • How to use discriminant to identify the nature of roots
  • How to convert word problems into quadratic equations

1. Standard Form of a Quadratic Equation

The standard form is:

ax^2 + bx + c = 0

where:

  • a, b, and c are real numbers
  • a != 0
  • x is the variable

Examples

EquationIs it quadratic?Reason
x^2 + 5x + 6 = 0YesHighest power is 2
3x^2 - 7 = 0YesHighest power is 2
4x + 9 = 0NoHighest power is 1
x^3 + x + 1 = 0NoHighest power is 3

2. Roots of a Quadratic Equation

A value of x that satisfies the equation is called a root or solution.

Example:

x^2 - 5x + 6 = 0

Try x = 2:

2^2 - 5(2) + 6 = 4 - 10 + 6 = 0

So, 2 is a root.

Try x = 3:

3^2 - 5(3) + 6 = 9 - 15 + 6 = 0

So, 3 is also a root.

3. Solving by Factorisation

This is the fastest method when factors are easy.

Steps

1. Write the equation in standard form. 2. Split the middle term. 3. Take common factors from pairs. 4. Factorise completely. 5. Put each factor equal to zero.

Solved Example

Solve:

x^2 + 7x + 12 = 0

Find two numbers whose product is 12 and sum is 7.

They are 3 and 4.

x^2 + 3x + 4x + 12 = 0 x(x + 3) + 4(x + 3) = 0 (x + 3)(x + 4) = 0

So:

x + 3 = 0 or x + 4 = 0 x = -3 or x = -4
Answerx = -3, -4.

4. Quadratic Formula

For:

ax^2 + bx + c = 0

the roots are:

x = (-b +/- sqrt(b^2 - 4ac)) / 2a

This method works for all quadratic equations where real roots exist.

Solved Example

Solve:

2x^2 - 5x + 3 = 0

Here:

a = 2, b = -5, c = 3

Apply formula:

x = (-(-5) +/- sqrt((-5)^2 - 4(2)(3))) / 2(2) x = (5 +/- sqrt(25 - 24)) / 4 x = (5 +/- 1) / 4

So:

x = 6/4 = 3/2 x = 4/4 = 1
Answerx = 1, 3/2.

5. Discriminant and Nature of Roots

The expression:

D = b^2 - 4ac

is called the discriminant.

DiscriminantNature of Roots
D > 0Two distinct real roots
D = 0Two equal real roots
D < 0No real roots

Example

Find the nature of roots of:

x^2 - 4x + 4 = 0

Here:

a = 1, b = -4, c = 4 D = b^2 - 4ac D = (-4)^2 - 4(1)(4) D = 16 - 16 D = 0

So the roots are real and equal.

6. Completing the Square

This method converts a quadratic expression into a perfect square form.

Example

Solve:

x^2 + 6x + 5 = 0

Move constant:

x^2 + 6x = -5

Take half of coefficient of x, square it, and add to both sides.

Half of 6 is 3, and 3^2 = 9.

x^2 + 6x + 9 = -5 + 9 (x + 3)^2 = 4 x + 3 = +/- 2

So:

x = -1 or x = -5

7. Word Problems

Word problems are the most important part of this chapter. The marks are in equation formation.

Method

1. Read what has to be found. 2. Let the unknown be x. 3. Convert the condition into an equation. 4. Bring it to standard form. 5. Solve. 6. Reject impossible values if the question is about age, length, speed, or number of objects.

Example: Consecutive Integers

The product of two consecutive positive integers is 306. Find the integers.

Let the integers be:

x and x + 1

Given:

x(x + 1) = 306 x^2 + x - 306 = 0

Factorise:

x^2 + 18x - 17x - 306 = 0 x(x + 18) - 17(x + 18) = 0 (x + 18)(x - 17) = 0

So:

x = -18 or x = 17

Since positive integers are required, reject -18.

Answer17 and 18.

8. Important Question Types

Question TypeBest Method
Simple equation with easy factorsFactorisation
Equation with difficult factorsQuadratic formula
Nature of rootsDiscriminant
Form equation from storyDefine variable first
Equal roots conditionPut D = 0
No real roots conditionShow D < 0

9. Practice Questions

1. Solve x^2 - 8x + 15 = 0. 2. Solve 2x^2 + x - 6 = 0. 3. Find the nature of roots of 3x^2 - 2x + 1 = 0. 4. Find the value of k if x^2 + 4x + k = 0 has equal roots. 5. The product of two consecutive positive integers is 420. Find the integers. 6. The sum of squares of two consecutive odd positive integers is 394. Find them. 7. Solve 4x^2 - 4ax + (a^2 - b^2) = 0.

10. Quick Answers

1. x = 3, 5. 2. x = 3/2, -2. 3. D = -8, so no real roots. 4. D = 0; 16 - 4k = 0; k = 4. 5. 20 and 21. 6. 13 and 15. 7. x = (a + b)/2, (a - b)/2.

Common Mistakes to Avoid

  • Forgetting that a cannot be zero in a quadratic equation.
  • Splitting the middle term with wrong signs.
  • Writing b^2 - 4ac incorrectly when b is negative.
  • Forgetting +/- in the quadratic formula.
  • Not rejecting negative answers in real-life word problems.
  • Giving only roots without showing the equation formation.

Final Revision Checklist

  • I can identify a quadratic equation.
  • I can solve by factorisation.
  • I can apply the quadratic formula correctly.
  • I can find the discriminant and nature of roots.
  • I can form equations from word problems.
  • I can reject impossible values in application questions.

Internal Links

  • [CBSE Class 10 Maths guide](/school/cbse/class/10/maths)
  • [Real Numbers notes](/school/cbse/class/10/maths/cbse-real-numbers)
  • [Practice aptitude questions](/aptitude)

Frequently Asked Questions

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to 0.

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