I’m okay expanding something like a number times a bracket, but as soon as a minus sign shows up I start second-guessing everything. For example, with 3(x + 4) – 2(x – 5), I keep forgetting whether that -2 should hit both terms inside the second bracket. Same with things like -(x – 7) or 1 – (x + 2) – do I flip both signs inside, or just the first one, or am I overthinking it? Is there a simple way to keep track of the signs so I don’t mess it up every other line?
Also, squared brackets melt my brain a bit. With (x + 3)^2, I know it’s not just x^2 + 9, but my brain really wants it to be. And when it’s mixed like (2x – 3)(x + 4), I lose track of which bits multiply and what the signs should be. Do you have any easy-to-remember tricks (like an area/box method or a quick mental checklist) to expand these without relying on me getting lucky? Bonus points if there’s a way to spot patterns like (a – b)(a + b) without memorizing a giant list.
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5 Responses
Distribute to every term in the bracket (treat −(…) as −1×(…)); for squares, write (a+b)^2 as (a+b)(a+b), multiply each term by each term, use the sign rules (+×+=+, +×−=−, −×−=+), then collect like terms; useful patterns: (a−b)(a+b)=a^2−b^2 and (a±b)^2=a^2±2ab+b^2.
Example: 3(x+4) − 2(x−5) = 3x+12 − 2x+10 = (3x−2x) + (12+10) = x+22.
Think of every bracket as being multiplied by whatever is stuck to its left, and treat a minus as a sneaky “times −1.” That one idea cleans up most sign snafus: −(x − 7) = (−1)(x − 7) = −x + 7, and 1 − (x + 2) = 1 + (−1)(x + 2) = 1 − x − 2 = −x − 1. For your example, 3(x + 4) − 2(x − 5), the −2 must hit both terms: 3x + 12 − 2x + 10 = x + 22. When two brackets multiply, use “each-with-each” (FOIL if you like) or a quick 2×2 box: put 2x and −3 on top, x and 4 on the side, fill the four boxes (2x·x=2x^2, 2x·4=8x, −3·x=−3x, −3·4=−12), then add them: 2x^2 + 5x − 12. For squares, (a ± b)^2 is not just a^2 + b^2; it’s a^2 ± 2ab + b^2, so (x + 3)^2 = x^2 + 6x + 9. A lovely pattern to spot is conjugates: (a − b)(a + b) = a^2 − b^2 (the middle terms cancel). Quick checklist before you move on: 1) distribute to every term, 2) keep the “times −1” idea for lone minuses, 3) combine like terms at the end, 4) for squares, remember the 2ab middle term, and 5) for mixed products, box or FOIL and then tidy. Your brain can relax-no luck required, just these little habits.
Ooh, signs and squares are where the sneaky goblins live! The way I keep myself sane (I think this is right, though I sometimes second-guess it) is to treat anything stuck to a bracket as multiplying the whole bracket-there’s a “hidden number” glued on. So 3(x + 4) − 2(x − 5): the −2 multiplies both terms, giving 3x + 12 − 2x + 10, which tidies to x + 22. Same trick for a lone minus: -(x − 7) is really (−1)(x − 7) = −x + 7; and 1 − (x + 2) = 1 − x − 2 = −x − 1 after you distribute the minus, then combine. For squares, I remind myself that (x + 3)^2 means (x + 3)(x + 3), so you get x^2 + 3x + 3x + 9 = x^2 + 6x + 9 (I’m pretty sure the middle “double” term 2·x·3 is the bit people forget). With mixed products like (2x − 3)(x + 4), I do two mini-distributions: 2x hits x and 4 (2x^2 + 8x), then −3 hits x and 4 (−3x − 12), combine to 2x^2 + 5x − 12. If signs trip you up, a quick 2×2 box helps: put 2x and −3 across the top, x and 4 down the side, fill each cell by multiplying, then add everything-seeing the four pieces makes the sign pattern pop. Pattern-spotting bonus: (a − b)(a + b) is a “difference of squares” because the cross terms cancel (ab and −ab), leaving a^2 − b^2-no memorization, just symmetry magic.
Rule of thumb: a multiplier (including a “−” which is just −1) distributes to every term in the bracket, then you combine like terms; for products/squares multiply every term by every term, so (x+a)² = x² + 2ax + a² and (a−b)(a+b) = a² − b².
Example: 3(x+4) − 2(x−5) = 3x+12 −2x+10 = x+22; (2x−3)(x+4) = 2x²+8x−3x−12 = 2x²+5x−12; (x+3)² = x²+6x+9.
A simple rule: always distribute to every term, carrying the sign-so 3(x+4)−2(x−5)=3x+12−2x+10, −(x−7)=−x+7, and 1−(x+2)=1−x−2. For products/squares, use a 2×2 box so each term multiplies once-(x+3)^2=x^2+6x+9, (2x−3)(x+4)=2x^2+5x−12, and spot (a−b)(a+b)=a^2−b^2; quick guide: https://www.khanacademy.org/math/algebra/polynomial-factorization/expanding-products-of-binomials/a/multiplying-binomials-does the sign distribution or the box step trip you up more?