I keep tripping over substitution and minus signs. If x = -3, I get confused about expressions like -x^2, (-x)^2, 5 – x^2, and (2 – x)^2 – which parts are actually being squared, and when does the negative sign get squared with it? I thought I understood order of operations, but the little dash in front of x^2 keeps tricking me.
I also get stuck when the thing I’m substituting is an expression. For example, if x = 2k – 5 and the original expression is -3x^2 + 4x – 7, am I supposed to write -3(2k – 5)^2 + 4(2k – 5) – 7 every time, or is that overkill? And what about something like 10 – x when x = 3k – 1 – do I need 10 – (3k – 1), or is 10 – 3k – 1 okay?
Is there a simple way to think about when parentheses are required during substitution so I stop losing minus signs? Any help appreciated!
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3 Responses
Quick rule: exponents hit the nearest thing only, so a leading minus isn’t squared; that’s why with x = -3 we get -x^2 = -(x^2) = -9 but (-x)^2 = 9, and when plugging in expressions you should always drop them in with parentheses first: -3x^2 + 4x – 7 becomes -3(2k – 5)^2 + 4(2k – 5) – 7, and 10 – x with x = 3k – 1 is 10 – (3k – 1), not 10 – 3k – 1. I used to lose minus signs all the time until a teacher told me to treat whatever I substitute as a single “blob” in parentheses-then apply powers/negatives-and it felt like unlocking an order-of-operations superpower!
Parentheses are the seatbelts of substitution-always buckle whatever you plug in (especially negatives or multi-term expressions); exponents go first, so -x^2 means -(x^2) while (-x)^2 squares the sign too. Example sprint: if x = -3 then -x^2 = -(-3)^2 = -9, (-x)^2 = (-(-3))^2 = 9, 5 – x^2 = 5 – (-3)^2 = -4, (2 – x)^2 = (2 – (-3))^2 = 25; if x = 2k – 5, then -3x^2 + 4x – 7 = -3(2k – 5)^2 + 4(2k – 5) – 7; if x = 3k – 1, then 10 – x = 10 – (3k – 1) = 11 – 3k (not 10 – 3k – 1).
Oh, the sneaky little minus sign! It loves to play hide-and-seek. Short version first: if you want a rule you can trust every time, put the thing you’re substituting inside parentheses. Parentheses are seatbelts for expressions. They keep signs from flying out the window when you turn a sharp algebraic corner.
Let me break it down into two big ideas, then we’ll do your examples.
Big idea 1: What does the exponent “stick to”?
– An exponent only applies to the single thing immediately to its left.
– If you want the exponent to apply to more than one thing (like a whole expression), you must use parentheses.
So:
– x^2 means “square x.”
– (-x)^2 means “square the negative of x,” which is the same as x^2.
– -x^2 means “take x^2, then slap a minus on the front,” i.e., -(x^2).
– 5 − x^2 means 5 − (x^2).
– (2 − x)^2 means square the entire binomial (2 − x), so both the 2 and the x are involved.
Big idea 2: Substituting expressions? Parentheses, always.
– When you replace x with something that has a sign (like −3), or any operation (like 2k − 5), wrap it in parentheses everywhere x appears.
– After that, simplify using order of operations (expand, distribute, combine like terms).
Now your specific cases with x = −3
– -x^2: The exponent hits x first, then the leading minus. So x^2 = (−3)^2 = 9, and then the minus makes it −9.
– (-x)^2: First negate x, so −x = −(−3) = 3, then square: 3^2 = 9.
– 5 − x^2: This is 5 − (x^2) = 5 − (−3)^2 = 5 − 9 = −4.
– (2 − x)^2: 2 − (−3) = 5, then 5^2 = 25.
See how parentheses decide what the square “grabs”?
Substituting an expression: x = 2k − 5 into -3x^2 + 4x − 7
– Do the safe thing: write -3(2k − 5)^2 + 4(2k − 5) − 7. That’s not overkill; that’s correct.
– Now simplify:
– (2k − 5)^2 = 4k^2 − 20k + 25.
– Multiply by −3: −12k^2 + 60k − 75.
– 4(2k − 5) = 8k − 20.
– Combine everything: (−12k^2 + 60k − 75) + (8k − 20) − 7 = −12k^2 + 68k − 102.
Subtracting an expression: 10 − x when x = 3k − 1
– Start with 10 − (3k − 1). Those parentheses matter.
– Now distribute the minus across the whole parenthesis: 10 − 3k + 1 = 11 − 3k.
– If you wrote 10 − 3k − 1, that would be 9 − 3k, which is wrong. Quick check: let k = 0. Original gives 11; the “dropped parentheses” gives 9.
When can I skip parentheses?
– If you’re adding, A + (B) = A + B, so those parentheses don’t change anything.
– If there is multiplication or an exponent involved, parentheses are usually essential:
– 4x with x = 2k − 5 should be 4(2k − 5), not 4·2k − 5 (that would miss the −5).
– x^2 with x = −3 should be (−3)^2; if you write −3^2, most conventions interpret that as −(3^2).
– For plain positive numbers, parentheses are less crucial, but it’s still good habit. For negatives, they’re crucial around exponents: x^2 with x = −3 must be (−3)^2.
A tiny mental trick that saved me
I used to lose a minus sign on every other problem. My teacher drew a box for the variable and said, “First write the formula with a box: f(□) = -3(□)^2 + 4(□) − 7. Then drop your x-value or expression into the box with parentheses.” It clicked! Parentheses as “hugs” for whatever you’re plugging in. I started saying “plug in, then peel out”: plug the expression in with parentheses, then peel the parentheses off by distributing and simplifying. The day I stopped treating that leading minus as part of the square was the day my algebra errors shrank by half.
Quick summary rules to keep you safe
– Always replace x by (stuff). Parentheses first, simplify second.
– Exponent grabs only the one thing to its left; parentheses decide what that is.
– A leading minus is not part of the exponent: -x^2 = -(x^2), but (-x)^2 = x^2.
– Subtracting an expression flips every sign inside: A − (B − C) = A − B + C.
If you follow “wrap the substitution in parentheses,” you’ll stop losing those sneaky minus signs. It’s the single best habit here.