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Solving by factorisation

Factorisation means rewriting a quadratic equation as a product of two simpler expressions and then using the zero product principle.

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Student-friendly explanation

This method works best when the quadratic can be split into factors with neat numbers. After factorising, each factor is set to zero, which gives the roots quickly.

How to write this in exams

  1. 1

    Start with the exact idea

    Factorisation means rewriting a quadratic equation as a product of two simpler expressions and then using the zero product principle.

  2. 2

    Then show how to use it

    1. Write the equation in standard form. 2. Find two numbers with the correct product and sum. 3. Split the middle term or directly factorise. 4. Set each factor equal to zero. 5. Write the roots.

  3. 3

    Add one concrete example

    x^2 - 7x + 12 = 0 becomes (x - 3)(x - 4) = 0.

  4. 4

    Avoid this incomplete answer

    Writing (x - 2)(x - 10) for x^2 - 12x + 20 = 0 is wrong because the sum does not match -12.

Definition

Factorisation means rewriting a quadratic equation as a product of two simpler expressions and then using the zero product principle.

Example

x^2 - 7x + 12 = 0 becomes (x - 3)(x - 4) = 0.

Rule to remember

If ax^2 + bx + c = (px + q)(rx + s), then solve by setting each factor equal to zero. Use the zero product principle after factorisation.

Memory hook

Find the pair, split the middle, solve the two brackets.

Examples and method

Worked example

Solve x^2 - 9x + 20 = 0. We need two numbers whose product is 20 and sum is -9, which are -4 and -5. So x^2 - 9x + 20 = (x - 4)(x - 5) = 0, giving x = 4 or x = 5.

Method to apply

1. Write the equation in standard form. 2. Find two numbers with the correct product and sum. 3. Split the middle term or directly factorise. 4. Set each factor equal to zero. 5. Write the roots.

Diagram support

A factor tree or bracket split can help students see the two factors before setting them equal to zero.

How CBSE asks it

CBSE often asks for solving by factorisation when the quadratic has simple factors and neat roots.

Avoid common mistakes

Common confusion

Students often choose the wrong factor pair because they match only the product and ignore the middle term.

Common wrong answer

Writing (x - 2)(x - 10) for x^2 - 12x + 20 = 0 is wrong because the sum does not match -12.

Exam tip

Check both the product and the sum while choosing factor pairs, especially for negative middle terms.

Quick check

Why does factorisation help in solving a quadratic equation quickly?

Because it turns one quadratic equation into two simple linear factors. Then each factor can be set equal to zero, which gives the roots without using a long formula.

Answer writing and exam use

1-mark answer

Factorisation means rewriting a quadratic equation as a product of two simpler expressions and then using the zero product principle.

2-mark answer

Factorisation means rewriting a quadratic equation as a product of two simpler expressions and then using the zero product principle. If ax^2 + bx + c = (px + q)(rx + s), then solve by setting each factor equal to zero. Use the zero product principle after factorisation. x^2 - 7x + 12 = 0 becomes (x - 3)(x - 4) = 0.

3-mark answer

This method works best when the quadratic can be split into factors with neat numbers. After factorising, each factor is set to zero, which gives the roots quickly. If ax^2 + bx + c = (px + q)(rx + s), then solve by setting each factor equal to zero. Use the zero product principle after factorisation. Solve x^2 - 9x + 20 = 0. We need two numbers whose product is 20 and sum is -9, which are -4 and -5. So x^2 - 9x + 20 = (x - 4)(x - 5) = 0, giving x = 4 or x = 5. CBSE often asks for solving by factorisation when the quadratic has simple factors and neat roots. Writing (x - 2)(x - 10) for x^2 - 12x + 20 = 0 is wrong because the sum does not match -12.
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