I’m prepping for a test and my brain keeps doing cartwheels when I try to simplify expressions with parentheses and minus signs. I keep thinking, “Just smoosh the like things together!” and then… chaos.
Example I’m stuck on: Simplify 3x(2x – 5) + 4(1 – x) – (x^2 – 9).
My totally wrong attempt (please tell me why each part is bad):
– 3x(2x – 5) = 6x – 5
– 4(1 – x) = 4 – x
– -(x^2 – 9) = -x^2 – 9
Then I combined everything and got -x^2 + 5x – 10. I’m pretty sure that’s nonsense, but I can’t seem to see exactly which rule(s) I’m breaking at each step. Can someone point out the specific mistakes and how to think about them so I stop doing this?
Follow-up questions:
– Is there a “best order” to handle expressions like this? Should I always distribute everything first and then combine, or is it sometimes smarter to factor (like x^2 – 9) before doing anything?
– As a quick check, if I plug in a number (like x = 1) to compare my result to the original, is that a reliable way to catch mistakes, or can that be misleading?
– Also, I keep wanting to cancel an x from x(2x – 5) to magically get (2 – 5). Am I ever allowed to do that, or is that always illegal brain-gremlin behavior?
Thank you! I’m trying to lock this down before my test and my scratch paper is starting to look like a tangle of spaghetti.
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3 Responses
I feel this! Parentheses plus minus signs can turn into a bowl of algebra spaghetti fast. The safest routine is “distribute, then combine.” Think of distribution like handing out a sticker to every kid in a line: nobody gets skipped, and everyone gets the same sticker. For your example, 3x(2x – 5) should be 3x·2x + 3x·(–5) = 6x^2 – 15x; your slip was dropping one x, which turns 6x^2 into 6x. Next, 4(1 – x) is 4·1 + 4·(–x) = 4 – 4x; you gave the 4 to the 1 but not to the x. Finally, –(x^2 – 9) means multiply by –1, flipping both signs: –x^2 + 9, not –x^2 – 9. Now combine like terms: (6x^2 – x^2) + (–15x – 4x) + (4 + 9) = 5x^2 – 19x + 13. Quick plug-in check at x = 1: original gives –1, and 5(1)^2 – 19(1) + 13 is also –1, so the simplification passes the vibe check.
On “best order,” distributing first and then combining like terms is the most reliable default. Factoring can be smart when it creates a common factor or you’re in a fraction and can cancel, but here factoring x^2 – 9 into (x – 3)(x + 3) doesn’t actually help with the rest of the sum. As a check, plugging in values is great for catching mistakes; one value can coincidentally agree, so try two or three different numbers if you want more confidence. And about “canceling an x” from x(2x – 5): that’s the brain gremlin talking. You can only cancel common factors in a product when everything is being multiplied and you’re dividing the whole expression by that factor. Inside 2x – 5 there’s subtraction, so no canceling. A quick refresher on the distributive property and combining like terms: https://www.khanacademy.org/math/algebra-basics/algebra-basics-polynomials/algebra-basics-distributive-property/a/distributive-property-review
Your slips are all distribution/sign issues: 3x(2x−5)=6x^2−15x (you have to hit both terms and the x makes x^2), 4(1−x)=4−4x, and a leading minus flips signs: −(x^2−9)=−x^2+9; combine to get 5x^2−19x+13, and a quick check x=1 gives −1 both ways. Best order: distribute then combine (factoring first is fine if it clearly helps), plugging in a number is a good sniff test not a proof, and you can only “cancel an x” if every term shares that factor after factoring-never across a plus/minus.
You’re running into three classic traps: forgetting to distribute to every term, forgetting that x·x gives x², and forgetting that a leading minus flips signs inside parentheses. Let’s fix your example carefully. Distribute 3x across both terms: 3x(2x − 5) = 6x² − 15x (your “6x − 5” missed the x·x and the −15x). Next, 4(1 − x) = 4 − 4x (you wrote 4 − x, but the x must also be multiplied by 4). Finally, a minus in front of parentheses means multiply by −1: −(x² − 9) = −x² + 9 (you had both terms negative). Now collect like terms only (x² with x², x with x, constants with constants): 6x² − 15x + 4 − 4x − x² + 9 = 5x² − 19x + 13. Quick check with x = 1: original gives 3·1·(2 − 5) + 4(1 − 1) − (1 − 9) = −9 + 0 + 8 = −1; your result gave −6, while 5x² − 19x + 13 gives 5 − 19 + 13 = −1, so it matches. About order: a reliable habit is “remove parentheses first, then combine like terms,” treating a leading minus as −1 times the bracket; factoring x² − 9 into (x − 3)(x + 3) doesn’t help here because nothing cancels and you’d just re-expand. Plugging numbers is a good error detector (one mismatch proves it’s wrong), but matching at a few values is not a proof; still, it’s a solid sanity check. And about “canceling an x”: cancellation works only with common factors in a product or a fraction, never across addition/subtraction; for instance, [x(2x − 5)]/x = 2x − 5 for x ≠ 0, but x(2x − 5) alone can’t be “reduced” to 2 − 5. Tiny practice example: 2y(3y + 1) − (y² − 4) + 5(2 − y) → 6y² + 2y − y² + 4 + 10 − 5y → 5y² − 3y + 14.