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Roots of a quadratic equation

The roots of a quadratic equation are the values of x that make the equation true, meaning they make the left side equal to zero.

In one line: A root is a value that makes the equation disappear into zero.

Practice This Concept

Understand it

A root is not just any number; it is a value that satisfies the equation exactly. When we solve a quadratic equation, we are looking for these values only.

Example

For x² - 5x + 6 = 0, the roots are 2 and 3.

Formula and rule

If α and β are the roots of ax² + bx + c = 0, then the equation can be written as a(x - α)(x - β) = 0.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

In x² - 5x + 6 = 0, factorisation gives (x - 2)(x - 3) = 0.

1So x = 2 or x = 3.
2Substituting each value makes the left side zero, so both are roots.

Method you can reuse

1Write the equation in standard form.
2Solve it using factorisation, formula, or completing the square.
3The obtained values are the roots.
4Verify by substitution when asked.

Visual tip

A number line can help show the root values as points where the equation is satisfied.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What does it mean when x = 3 is a root of a quadratic equation?

Think first, then show answer

It means that when 3 is substituted into the equation, the left side becomes zero. So x = 3 satisfies the equation exactly and is one solution.

Common wrong answer

Saying 5 is a root of x² - 5x + 6 = 0.

Why it loses marks: Substitution does not make the equation true.

Don't mix this up

Students sometimes call any number written in the equation a root, even if it does not satisfy the equation.

Exam tip

To check a root, substitute it back into the original equation and see whether the left side becomes zero.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions may ask students to identify roots directly, verify a proposed root, or match roots with an equation.

1
Enough for 1 mark
The roots of a quadratic equation are the values of x that make the equation true, meaning they make the left side equal to zero.
2
Add for 2 marks
Example: For x² - 5x + 6 = 0, the roots are 2 and 3.
3
Add for 3 marks
A root is not just any number; it is a value that satisfies the equation exactly.

MCQ Quiz

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