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Common sign errors in factorisation

These are mistakes where students choose the wrong signs in the factors while factorising a quadratic equation.

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Student-friendly explanation

Sign errors happen because students often match only the product and forget how the middle term controls the signs. Correct factorisation must satisfy both the constant term and the middle term.

How to write this in exams

  1. 1

    Start with the exact idea

    These are mistakes where students choose the wrong signs in the factors while factorising a quadratic equation.

  2. 2

    Then show how to use it

    1. Find two numbers with the correct product. 2. Check their sum against the middle coefficient. 3. Decide the signs from the middle term. 4. Expand quickly if needed to verify.

  3. 3

    Add one concrete example

    x^2 - 8x + 15 = (x - 3)(x - 5), not (x + 3)(x + 5).

  4. 4

    Avoid this incomplete answer

    Using (x + 3)(x + 5) for x^2 - 8x + 15 is wrong because the expansion gives +8x, not -8x.

Definition

These are mistakes where students choose the wrong signs in the factors while factorising a quadratic equation.

Example

x^2 - 8x + 15 = (x - 3)(x - 5), not (x + 3)(x + 5).

Rule to remember

Check the product of constants and the sum of middle terms carefully while factorising a quadratic.

Memory hook

Product fixes the pair, middle term fixes the signs.

Examples and method

Worked example

For x^2 - 8x + 15, the numbers 3 and 5 multiply to 15 and add to 8. Because the middle term is negative, the correct factorisation is (x - 3)(x - 5).

Method to apply

1. Find two numbers with the correct product. 2. Check their sum against the middle coefficient. 3. Decide the signs from the middle term. 4. Expand quickly if needed to verify.

Diagram support

A sign-check table for product and sum helps prevent bracket-sign confusion.

How CBSE asks it

Students may be asked to identify the correct factorisation or spot the sign mistake in a worked solution.

Avoid common mistakes

Common confusion

Students may use two positive factors even when the middle term is negative.

Common wrong answer

Using (x + 3)(x + 5) for x^2 - 8x + 15 is wrong because the expansion gives +8x, not -8x.

Exam tip

For a positive constant with a negative middle term, both signs inside the brackets are usually negative.

Quick check

Why is (x + 4)(x + 5) wrong for x^2 - 9x + 20?

Because it gives a positive middle term, not -9x. The correct factorisation must produce both the product 20 and the sum -9, so the signs must be negative.

Answer writing and exam use

1-mark answer

These are mistakes where students choose the wrong signs in the factors while factorising a quadratic equation.

2-mark answer

These are mistakes where students choose the wrong signs in the factors while factorising a quadratic equation. Check the product of constants and the sum of middle terms carefully while factorising a quadratic. x^2 - 8x + 15 = (x - 3)(x - 5), not (x + 3)(x + 5).

3-mark answer

Sign errors happen because students often match only the product and forget how the middle term controls the signs. Correct factorisation must satisfy both the constant term and the middle term. Check the product of constants and the sum of middle terms carefully while factorising a quadratic. For x^2 - 8x + 15, the numbers 3 and 5 multiply to 15 and add to 8. Because the middle term is negative, the correct factorisation is (x - 3)(x - 5). Students may be asked to identify the correct factorisation or spot the sign mistake in a worked solution. Using (x + 3)(x + 5) for x^2 - 8x + 15 is wrong because the expansion gives +8x, not -8x.
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