CBSE Class 10 Maths Revision Notes Chapter 4 Quadratic Equations 2026–27

Quadratic Equations Class 10 explains equations in which the highest power of the variable is 2.
For CBSE Class 10 Maths, this chapter covers standard form, roots, solution methods and nature of roots.

Quadratic Equations are formed when a quadratic polynomial is equated to zero. In Chapter 4, students learn how to write these equations in standard form, find their roots and understand what the roots tell us.

Use these CBSE Class 10 Maths Revision Notes Chapter 4 to revise the chapter step by step. Start with the standard form, then move to factorisation, completing the square, quadratic formula, discriminant and word problems.

Key Takeaways

• Standard form: ax² + bx + c = 0, where a ≠ 0.
• Root: A value of x that satisfies the quadratic equation.
• Quadratic formula: x = (-b ± √(b² - 4ac))/(2a).
• Discriminant: D = b² - 4ac decides the nature of roots.

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CBSE Class 10 Maths Chapter 4 Quadratic Equations at a Glance

Class 10 Maths Chapter 4 Quadratic Equations connects polynomials with equations and their roots. It also shows how real-life situations involving area, speed, time and numbers can lead to quadratic equations.

Standard Form of Quadratic Equation for Class 10

The standard form of quadratic equation is the first formula students use in this chapter. It helps identify the values of a, b and c before solving.

Copyable format:

ax² + bx + c = 0

a, b and c are real numbers.

a ≠ 0

Here:

• ax² is the quadratic term.
• bx is the linear term.
• c is the constant term.
• a cannot be zero because the equation would become linear.

Examples of quadratic equations:

• 2x² + x - 300 = 0
• 2x² - 3x + 1 = 0
• x² - 6x + 8 = 0

How to Identify a Quadratic Equation

Some equations do not look quadratic at first. Simplify the equation first, then check whether it can be written as ax² + bx + c = 0.

Example:

(x - 2)² + 1 = 2x - 3

Expand the left side:

(x - 2)² + 1 = x² - 4x + 4 + 1

(x - 2)² + 1 = x² - 4x + 5

Now write the equation:

x² - 4x + 5 = 2x - 3

Bring all terms to one side:

x² - 4x - 2x + 5 + 3 = 0

x² - 6x + 8 = 0

This is a quadratic equation because it is in the form:

ax² + bx + c = 0

Roots of Quadratic Equation in Class 10 Maths

The roots of quadratic equation are the values of x that satisfy the equation. These roots are also the zeroes of the related quadratic polynomial.

For:

ax² + bx + c = 0

A real number α is a root if:

aα² + bα + c = 0

A quadratic equation can have at most two roots.

Example:

2x² - 3x + 1 = 0

Check x = 1:

2(1)² - 3(1) + 1 = 0

2 - 3 + 1 = 0

0 = 0

So, x = 1 is a root.

Copyable format:

If aα² + bα + c = 0, then α is a root of ax² + bx + c = 0.

Solving Quadratic Equations by Factorisation Method

The factorisation method writes the quadratic expression as a product of two linear factors. After factorisation, each factor is equated to zero.

Steps:

• Write the equation in standard form.
• Split the middle term.
• Factorise by grouping.
• Equate each factor to zero.
• Find the roots.

Example:

Find the roots of:

2x² - 5x + 3 = 0

Split the middle term:

2x² - 5x + 3 = 2x² - 2x - 3x + 3

Group the terms:

2x² - 2x - 3x + 3 = 2x(x - 1) - 3(x - 1)

Take common factor:

2x(x - 1) - 3(x - 1) = (x - 1)(2x - 3)

So:

2x² - 5x + 3 = 0

(x - 1)(2x - 3) = 0

Now:

x - 1 = 0 or 2x - 3 = 0

x = 1 or x = 3/2

Answer:

Roots are 1 and 3/2.

Solving Quadratic Equations by Completing the Square Method

The completing the square method changes the quadratic expression into a perfect square form. This method is also used to understand how the quadratic formula is obtained.

Example:

Solve:

2x² - 5x + 3 = 0

Divide by 2:

x² - (5/2)x + 3/2 = 0

Move the constant term:

x² - (5/2)x = -3/2

Half of coefficient of x:

(5/2) ÷ 2 = 5/4

Add square of 5/4 on both sides:

x² - (5/2)x + (5/4)² = -3/2 + (5/4)²

Write as a square:

(x - 5/4)² = -3/2 + 25/16

(x - 5/4)² = -24/16 + 25/16

(x - 5/4)² = 1/16

Take square root:

x - 5/4 = ±1/4

So:

x = 5/4 ± 1/4

x = 3/2 or x = 1

Answer:

Roots are 1 and 3/2.

Quadratic Formula for Class 10 Maths Chapter 4

The quadratic formula gives the roots directly when the equation is in standard form. It is useful when factorisation is difficult.

For:

ax² + bx + c = 0, where a ≠ 0

The roots are:

x = (-b ± √(b² - 4ac))/(2a)

Copyable format:

x = (-b + √(b² - 4ac))/(2a)

x = (-b - √(b² - 4ac))/(2a)

The expression under the square root is called the discriminant.

D = b² - 4ac

So, the formula can also be written as:

x = (-b ± √D)/(2a)

Discriminant and Nature of Roots

The discriminant decides whether the roots are real and distinct, real and equal, or not real. This is one of the most important formula-based parts of CBSE Notes Class 10 Maths Chapter 4.

For:

ax² + bx + c = 0

Discriminant:

D = b² - 4ac

Copyable format:

If D > 0, roots are real and distinct.

If D = 0, roots are real and equal.

If D < 0, there are no real roots.

For D = 0:

x = -b/(2a), -b/(2a)

Relationship Between Roots and Coefficients

The relationship between roots and coefficients helps students find the sum and product of roots without solving the full equation.

For the quadratic equation:

ax² + bx + c = 0

Let α and β be the roots.

Then:

α + β = -b/a

αβ = c/a

Copyable format:

Sum of roots = α + β = -b/a

Product of roots = αβ = c/a

A quadratic equation can also be formed from roots.

If roots are α and β, then:

x² - (α + β)x + αβ = 0

Copyable format:

Quadratic equation from roots = x² - (sum of roots)x + product of roots = 0

Quadratic Equation Word Problems for Class 10

Quadratic equation word problems usually come from area, age, speed, distance, time and consecutive-number situations. The first step is to define the variable clearly.

Use these common forms:

Time = Distance/Speed

Area of rectangle = Length × Breadth

Consecutive positive integers = x and x + 1

Consecutive even integers = x and x + 2

Consecutive odd integers = x and x + 2

Example:

The product of two consecutive positive integers is 306.

Let the integers be:

x and x + 1

Then:

x(x + 1) = 306

x² + x = 306

x² + x - 306 = 0

This is the required quadratic equation.

Formula Table for CBSE Class 10 Maths Revision Notes Chapter 4

This table can be used with Class 10 Mathematics Revision Notes Chapter 4 for quick formula recall before solving textbook questions.

Solved Examples from Chapter 4 Quadratic Equations

Solved examples help students revise the correct method. Start with factorisation, then use the discriminant or quadratic formula when factors are not easy.

Example 1: Check Whether an Equation is Quadratic

Check whether the equation is quadratic:

(x - 2)² + 1 = 2x - 3

Expand:

x² - 4x + 4 + 1 = 2x - 3

x² - 4x + 5 = 2x - 3

Bring all terms to one side:

x² - 6x + 8 = 0

Answer:

The equation is quadratic because it is in the form ax² + bx + c = 0.

Example 2: Find Roots by Factorisation

Find the roots of:

6x² - x - 2 = 0

Split the middle term:

6x² - x - 2 = 6x² + 3x - 4x - 2

Group the terms:

6x² + 3x - 4x - 2 = 3x(2x + 1) - 2(2x + 1)

Take common factor:

3x(2x + 1) - 2(2x + 1) = (3x - 2)(2x + 1)

So:

(3x - 2)(2x + 1) = 0

Now:

3x - 2 = 0 or 2x + 1 = 0

x = 2/3 or x = -1/2

Answer:

Roots are 2/3 and -1/2.

Example 3: Find the Nature of Roots

Find the nature of roots of:

x² - 4x + 4 = 0

Compare with:

ax² + bx + c = 0

So:

a = 1

b = -4

c = 4

Discriminant:

D = b² - 4ac

D = (-4)² - 4(1)(4)

D = 16 - 16

D = 0

Answer:

The equation has two equal real roots.

Example 4: Form a Quadratic Equation from a Word Problem

The area of a rectangular plot is 528 m². Its length is one metre more than twice its breadth.

Let breadth be:

x m

Then length is:

2x + 1 m

Area:

Length × Breadth = 528

So:

x(2x + 1) = 528

2x² + x = 528

2x² + x - 528 = 0

Answer:

The required quadratic equation is 2x² + x - 528 = 0.

Chapter 4 Quadratic Equations Summary

These Class 10 Maths Chapter 4 Notes help students revise standard form, roots, factorisation method, completing the square method, quadratic formula, discriminant and nature of roots in one flow.

• A quadratic equation has standard form ax² + bx + c = 0.
• The condition for a quadratic equation is a ≠ 0.
• The roots are values of x that satisfy the equation.
• A quadratic equation can have at most two roots.
• Factorisation works by splitting the middle term.
• Completing the square converts the expression into a perfect square.
• The quadratic formula is:
  • x = (-b ± √(b² - 4ac))/(2a)
• The discriminant is:
  • D = b² - 4ac
• If D > 0, roots are real and distinct.
• If D = 0, roots are real and equal.
• If D < 0, there are no real roots.
• For roots α and β:
  • α + β = -b/a
  • αβ = c/a

Useful Important Questions Class 10 Maths Links