When the exam adds a minus sign, the rule from the 10-second trick doesn't change. You just make one sign check before you start.
Watch it backwards
Expand (x − 4)(x + 3) and you get x² + 3x − 4x − 12, which is x² − x − 12.
The rule still holds. −4 × 3 = −12, the last number. −4 + 3 = −1, the middle number. The signs just come along for the ride. (If you haven't seen it yet, Part 1 explains the rule.)
One check first: the sign of the last number
- Last number negative? Your two numbers have different signs: one plus, one minus.
- Last number positive? Both numbers take the sign of the middle number.
Worked example: x² − 2x − 15
The last number is negative, so the signs are different. The pairs for 15 are 1 and 15, and 3 and 5. Try the signs on 3 and 5: −5 + 3 = −2 ✓ (and 5 − 3 = +2, which is wrong).
So x² − 2x − 15 = (x − 5)(x + 3). Expand it to check: x² + 3x − 5x − 15 = x² − 2x − 15.
Factorise x² − 7x + 12.
Show the answer
The last number is positive and the middle is negative, so both numbers are negative. −3 × −4 = 12 and −3 + −4 = −7, so it's (x − 3)(x − 4). It's the same 3 and 4 as Part 1, just negative.
Save this for your next test
- Multiply to the last, add to the middle: the rule never changes.
- Check the sign of the last number first.
- Negative last number: different signs.
- Positive last number: both take the middle number's sign.
Read the video transcript
- Last time, factorising took ten seconds. Then the exam adds a minus sign.
- Good news: the rule doesn't change.
- Watch it backwards.
- x minus four, times x plus three.
- Multiply it out: x squared, minus x, minus twelve.
- Minus four times three is minus twelve. Minus four plus three is minus one.
- Same rule. The signs just come along for the ride.
- One check first: the sign of the last number.
- Negative? Your two numbers have different signs.
- Positive? They share the middle's sign.
- Try it: x squared, minus two x, minus fifteen.
- Last number negative, so different signs.
- Pairs for fifteen: one and fifteen. Three and five.
- Minus five plus three makes minus two. That's it!
- x minus five, times x plus three.
- Your turn: x squared, minus seven x, plus twelve. Three… two… one…
- Positive last, negative middle, so both negative.
- It's last time's three and four, just negative!
- x minus three, times x minus four.
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