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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    Saying 5 is a root of x^2 - 5x + 6 = 0 is wrong because substitution does not make the equation true.

Definition

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.

Example

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

Rule to remember

If α and β are the roots of ax^2 + bx + c = 0, then the equation can be written as a(x - α)(x - β) = 0.

Memory hook

A root is a value that makes the equation disappear into zero.

Examples and method

Worked example

In x^2 - 5x + 6 = 0, factorisation gives (x - 2)(x - 3) = 0. So x = 2 or x = 3. Substituting each value makes the left side zero, so both are roots.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

Saying 5 is a root of x^2 - 5x + 6 = 0 is wrong because substitution does not make the equation true.

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

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-mark answer

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. If α and β are the roots of ax^2 + bx + c = 0, then the equation can be written as a(x - α)(x - β) = 0. For x^2 - 5x + 6 = 0, the roots are 2 and 3.

3-mark answer

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. If α and β are the roots of ax^2 + bx + c = 0, then the equation can be written as a(x - α)(x - β) = 0. In x^2 - 5x + 6 = 0, factorisation gives (x - 2)(x - 3) = 0. So x = 2 or x = 3. Substituting each value makes the left side zero, so both are roots. Questions may ask students to identify roots directly, verify a proposed root, or match roots with an equation. Saying 5 is a root of x^2 - 5x + 6 = 0 is wrong because substitution does not make the equation true.
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