CBSE Class 10 Maths Quadratic Equations: Notes, Formulae and Quick Revision

Chapter-wise notes, formulae, methods, solved examples and mistake checks. Choose a topic, then revise the section you need.

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Quadratic Equations · Standard Form and Roots

Standard Form and Roots

Recognise a quadratic equation and understand what its roots represent before choosing a method.

Quick notes

Recognise the standard form

A quadratic equation can be written as $a x^{2} + b x + c = 0$ , where $a \neq = 0$ .

Know what the roots mean

The roots are the values of $x$ that make the quadratic expression equal to zero.

Formulae

Roots and coefficients

$α + β = - \frac{b}{a}$ , $α β = \frac{c}{a}$

Useful for forming or checking a quadratic equation.

Key methods

Check the equation first

Write the equation in standard form, identify $a$ , $b$ and $c$ , and then decide whether factorisation or the quadratic formula is more suitable.

Solved example

For $x^{2} - 5 x + 6 = 0$ , identify $a$ , $b$ and $c$ .

1.Compare with $a x^{2} + b x + c = 0$ .
2.So $a = 1$ , $b = - 5$ and $c = 6$ .

The negative sign belongs to $b$ , which matters in every later calculation.

Common mistakes

Writing $b = 5$ instead of $b = - 5$ .
Treating an equation as quadratic when the coefficient of $x^{2}$ is zero.

Practice

Practise Quadratic Equations Important Questions Competency Questions Previous Year Questions

Quadratic Equations · Solving Methods

Solving Methods

Choose factorisation when it is quick; use the quadratic formula when the factors are not obvious.

Quick notes

Factorisation is not guesswork

Look for two numbers with the required product and sum, then set each factor equal to zero.

Formula method needs signs

Before substituting, write $a$ , $b$ and $c$ separately with their signs.

Formulae

Quadratic formula

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

Use when factorisation is not quick or obvious.

Key methods

Factorisation

Write both factors clearly, then set each factor to zero.

Quadratic formula

Substitute carefully, simplify the discriminant, then check both roots.

Solved example

Solve $x^{2} - 5 x + 6 = 0$ .

1.Write $x^{2} - 5 x + 6 = (x - 2) (x - 3)$ .
2.Set each factor to zero: $x - 2 = 0$ or $x - 3 = 0$ .

Therefore, the roots are $x = 2$ and $x = 3$ .

Common mistakes

Forgetting that both factors must be set equal to zero.
Dropping the $\pm$ sign in the quadratic formula.

Practice

Practise Quadratic Equations Important Questions Competency Questions Previous Year Questions

Quadratic Equations · Nature of Roots

Nature of Roots

Use the discriminant to predict the roots before solving the full equation.

Quick notes

Read the discriminant

The discriminant tells you the nature of roots before you solve the equation fully.

Formulae

Discriminant

$D = b^{2} - 4 a c$

$D > 0$ : two real roots; $D = 0$ : equal roots; $D < 0$ : no real roots.

Key methods

Use $D$ before the formula

Identify $a$ , $b$ and $c$ , calculate $D$ , then state the nature of roots using the sign of $D$ .

Solved example

Find the nature of roots of $x^{2} - 6 x + 9 = 0$ .

1.Here $a = 1$ , $b = - 6$ , $c = 9$ .
2.Calculate $D = (- 6)^{2} - 4 (1) (9) = 0$ .

The equation has equal real roots.

Common mistakes

Using $b^{2} + 4 a c$ instead of $ $b^{2} - 4 a c$ .
Giving the roots instead of the requested nature of roots.

Practice

Practise Quadratic Equations Important Questions Competency Questions Previous Year Questions

Quadratic Equations · Situational Problems

Situational Problems

Turn the situation into an equation, solve it, then check that the answer fits the question.

Quick notes

Define the variable

State clearly what $x$ represents before forming the quadratic equation.

Formulae

Useful check

$a x^{2} + b x + c = 0$

After forming the equation, check every term against the information in the problem.

Key methods

Word-problem flow

Define the variable, translate the information into an equation, solve it, and reject any value that does not fit the situation.

Solved example

A solution gives two possible values for a length. What should you do next?

1.Substitute each value into the condition in the question.
2.Reject a negative or impossible length.

Keep only the value that makes sense in the stated situation.

Common mistakes

Not defining the variable.
Keeping an impossible negative value without checking the context.

Practice

Practise Quadratic Equations Important Questions Competency Questions Previous Year Questions

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CBSE 10

CBSE Class 10 Maths Quadratic Equations: Notes, Formulae and Quick Revision

Chapter-wise notes, formulae, methods, solved examples and mistake checks. Choose a topic, then revise the section you need.

Continue with question practice

CBQs Important Questions PYQs

ChapterTopic

Quick Concepts Revision replaces question-format filters with study tools for the selected chapter and topic.

Quadratic Equations · Standard Form and Roots

Standard Form and Roots

Recognise a quadratic equation and understand what its roots represent before choosing a method.

Quick notes

Recognise the standard form

A quadratic equation can be written as $a x^{2} + b x + c = 0$ , where $a \neq = 0$ .

Know what the roots mean

The roots are the values of $x$ that make the quadratic expression equal to zero.

Formulae

Roots and coefficients

$α + β = - \frac{b}{a}$ , $α β = \frac{c}{a}$

Useful for forming or checking a quadratic equation.

Key methods

Check the equation first

Write the equation in standard form, identify $a$ , $b$ and $c$ , and then decide whether factorisation or the quadratic formula is more suitable.

Solved example

For $x^{2} - 5 x + 6 = 0$ , identify $a$ , $b$ and $c$ .

1.Compare with $a x^{2} + b x + c = 0$ .
2.So $a = 1$ , $b = - 5$ and $c = 6$ .

The negative sign belongs to $b$ , which matters in every later calculation.

Common mistakes

Writing $b = 5$ instead of $b = - 5$ .
Treating an equation as quadratic when the coefficient of $x^{2}$ is zero.

Practice

Practise Quadratic Equations Important Questions Competency Questions Previous Year Questions

Quadratic Equations · Solving Methods

Solving Methods

Choose factorisation when it is quick; use the quadratic formula when the factors are not obvious.

Quick notes

Factorisation is not guesswork

Look for two numbers with the required product and sum, then set each factor equal to zero.

Formula method needs signs

Before substituting, write $a$ , $b$ and $c$ separately with their signs.

Formulae

Quadratic formula

$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

Use when factorisation is not quick or obvious.

Key methods

Factorisation

Write both factors clearly, then set each factor to zero.

Quadratic formula

Substitute carefully, simplify the discriminant, then check both roots.

Solved example

Solve $x^{2} - 5 x + 6 = 0$ .

1.Write $x^{2} - 5 x + 6 = (x - 2) (x - 3)$ .
2.Set each factor to zero: $x - 2 = 0$ or $x - 3 = 0$ .

Therefore, the roots are $x = 2$ and $x = 3$ .

Common mistakes

Forgetting that both factors must be set equal to zero.
Dropping the $\pm$ sign in the quadratic formula.

Practice

Practise Quadratic Equations Important Questions Competency Questions Previous Year Questions

Quadratic Equations · Nature of Roots

Nature of Roots

Use the discriminant to predict the roots before solving the full equation.

Quick notes

Read the discriminant

The discriminant tells you the nature of roots before you solve the equation fully.

Formulae

Discriminant

$D = b^{2} - 4 a c$

$D > 0$ : two real roots; $D = 0$ : equal roots; $D < 0$ : no real roots.

Key methods

Use $D$ before the formula

Identify $a$ , $b$ and $c$ , calculate $D$ , then state the nature of roots using the sign of $D$ .

Solved example

Find the nature of roots of $x^{2} - 6 x + 9 = 0$ .

1.Here $a = 1$ , $b = - 6$ , $c = 9$ .
2.Calculate $D = (- 6)^{2} - 4 (1) (9) = 0$ .

The equation has equal real roots.

Common mistakes

Using $b^{2} + 4 a c$ instead of $ $b^{2} - 4 a c$ .
Giving the roots instead of the requested nature of roots.

Practice

Practise Quadratic Equations Important Questions Competency Questions Previous Year Questions

Quadratic Equations · Situational Problems

Situational Problems

Turn the situation into an equation, solve it, then check that the answer fits the question.

Quick notes

Define the variable

State clearly what $x$ represents before forming the quadratic equation.

Formulae

Useful check

$a x^{2} + b x + c = 0$

After forming the equation, check every term against the information in the problem.

Key methods

Word-problem flow

Define the variable, translate the information into an equation, solve it, and reject any value that does not fit the situation.

Solved example

A solution gives two possible values for a length. What should you do next?

1.Substitute each value into the condition in the question.
2.Reject a negative or impossible length.

Keep only the value that makes sense in the stated situation.

Common mistakes

Not defining the variable.
Keeping an impossible negative value without checking the context.

Practice

Practise Quadratic Equations Important Questions Competency Questions Previous Year Questions