Factoring trinomials with negative numbers
Use this page when the signs are the part slowing you down, even though the basic factoring pattern already makes sense.
Start here if you want the short version before reading the full method.
- Negative numbers change the bracket signs, but the core checks stay the same: multiply to the constant term and add to the middle coefficient.
- The sign of the constant tells you whether the bracket signs match or differ.
What this topic means and what to look for first.
Many factorisation questions become harder only because the signs are less friendly, not because the structure is new.
A calm sign check usually solves more of the problem than trying a long list of random number pairs.
One reliable route through the topic.
- 1Write the trinomial clearly in standard form.
- 2Check the sign of the constant term first.
- 3If the constant is positive, the bracket signs match; if it is negative, the bracket signs differ.
- 4Choose a number pair that multiplies to the constant term and adds to the middle coefficient.
- 5Expand the brackets again to confirm the sign pattern really works.
See the method in action.
x^2 - 7x + 12
- The constant is positive, so the bracket signs will match.
- Because the middle term is negative, both signs should be negative.
- The pair 3 and 4 works, so the factorisation is (x - 3)(x - 4).
x^2 + x - 12
- The constant is negative, so the bracket signs must differ.
- Look for two numbers that multiply to -12 and add to 1.
- The pair 4 and -3 works, so the factorisation is (x + 4)(x - 3).
x^2 - 2x - 15
- The constant is negative, so one bracket sign is positive and the other is negative.
- The pair -5 and 3 multiplies to -15 and adds to -2.
- So the factorisation is (x - 5)(x + 3).
Things that commonly send the method off track.
- Seeing a negative middle term and assuming both bracket signs must be negative every time.
- Forgetting that a negative constant forces opposite bracket signs.
- Checking only the product and never checking the sum.
Use a short verification pass before moving on.
- Expand the brackets and pay special attention to the sign of the middle term.
- If the constant sign is wrong, revisit whether the brackets should match or differ.
Try a few variations before switching to a calculator or solver tool.
- x^2 - 9x + 20
- x^2 + 2x - 15
- x^2 - 11x + 24
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Short answers worth checking.
Use the same multiply-and-add checks as usual, but let the sign of the constant term guide whether the bracket signs match or differ.
A negative constant means one bracket sign should be positive and the other negative.
Continue with the next closely related topic.
Use the public site structure first, then switch into the solver tool only if you need a direct test.
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