medium importancemedium difficulty5 min

Common sign errors in factorisation

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

In one line: Product fixes the pair, middle term fixes the signs.

Understand it

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.

Example

x² - 8x + 15 = (x - 3)(x - 5), not (x + 3)(x + 5).

Formula and rule

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

See it worked out

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

Worked example

1For x² - 8x + 15, the numbers 3 and 5 multiply to 15 and add to 8.
2Because the middle term is negative, the correct factorisation is (x - 3)(x - 5).

Method you can reuse

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

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is (x + 4)(x + 5) wrong for x² - 9x + 20?

Think first, then show answer

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.

Common wrong answer

Using (x + 3)(x + 5) for x² - 8x + 15.

Why it loses marks: The expansion gives +8x, not -8x.

Don't mix this up

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

Exam tip

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

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

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

1
Enough for 1 mark
These are mistakes where students choose the wrong signs in the factors while factorising a quadratic equation.
2
Add for 2 marks
Example: x² - 8x + 15 = (x - 3)(x - 5), not (x + 3)(x + 5).
3
Add for 3 marks
Correct factorisation must satisfy both the constant term and the middle term.

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