The Minus Sign: 5 Classic Mistakes
Learn the 5 most common algebra errors so you never make them again.
The minus sign is small, but it is mighty! It is the single most common reason students lose marks in algebra exams. This lesson covers the 5 “classic” traps students fall into so you can spot them a mile away.
Mistake 1: Squaring a Negative Number
This is perhaps the most famous calculator error in history. There is a huge difference between squaring a negative number with brackets and without them.
The Rule
With Brackets: $(-3)^2$ means “negative three times negative three.”
$$(-3) [[times]] (-3) = 9$$
Without Brackets: $-3^2$ means “the negative version of three squared.” By order of operations (BIDMAS/PEMDAS), we square the 3 first, then apply the minus.
$$-(3 [[times]] 3) = -9$$
Key Takeaway: If you want to square a negative number, you must put it inside brackets.
Mistake 2: Substitution Without Protection
When you substitute a negative number into an algebraic expression, imagine the number is fragile. You must wrap it in “protective” brackets to keep it safe.
Example: If $x = -5$, evaluate $x^2 – 2x$.
❌ The Wrong Way (Lazy Substitution):
$-5^2 – 2-5$
This looks like “minus 5 squared minus 2 minus 5”. It is confusing and mathematically incorrect.
✅ The Right Way (Protective Brackets):
$(-5)^2 – 2(-5)$
Now we can calculate step-by-step:
1. Square the term: $(-5)^2 = 25$
2. Multiply the second part: $-2 [[times]] -5 = +10$
3. Add them: $25 + 10 = 35$
Mistake 3: The “Invisible -1”
When you see a minus sign directly outside a bracket, like $-(x + 3)$, it isn’t just a minus sign. It is actually an invisible $-1$ waiting to be multiplied.
The Expansion Rule:
$$-(a + b) = -1(a + b) = -a – b$$
Common Trap: Expanding $5 – 2(x – 4)$.
Many students write: $5 – 2x – 4$ (forgetting to multiply the second term).
Correct Method: Treat the multiplier as $-2$.
Step 1: $-2 [[times]] x = -2x$
Step 2: $-2 [[times]] -4 = +8$
Result: $5 – 2x + 8 = 13 – 2x$
Mistake 4: Subtracting a Negative
We often say “two minuses make a plus.” This is true, but only when they are touching (like $5 – -3$) or multiplying (like $-2 [[times]] -3$).
Watch out for this error:
$-5 – 3$
Some students see two minuses and think the answer is positive $8$. This is incorrect! Think of temperature:
If it is $-5^\circ$ and gets $3^\circ$ colder, it becomes $-8^\circ$.
Mistake 5: The Fraction Bar Trap
When a minus sign sits in front of a fraction, it applies to the entire numerator (top part), not just the first number.
Example: Simplify $5 – [[frac{x – 2}{3}]]$
You must treat the numerator $(x – 2)$ as if it has invisible brackets around it: $-(x – 2)$.
If we combine this into a single fraction, the minus distributes:
Numerator becomes: $-x + 2$ (because $- [[times]] -2 = +2$).
Practice Problems
Test your skills by fixing these common errors.
Problem 1: Evaluate $x^2$ when $x = -4$.
Show Answer
Step 1: Substitute with brackets.
$$(-4)^2$$
Step 2: Calculate.
$$-4 [[times]] -4 = 16$$
(Note: If you wrote $-4^2$, your calculator might give $-16$, which is wrong for this substitution!)
Problem 2: Expand and simplify: $10 - 3(x - 2)$.
Show Answer
Step 1: Identify the multiplier.
The multiplier is $-3$, not just $3$.
Step 2: Multiply the first term.
$$-3 [[times]] x = -3x$$
Step 3: Multiply the second term (Watch the signs!).
$$-3 [[times]] -2 = +6$$
Step 4: Combine with the 10.
$$10 - 3x + 6$$
Step 5: Simplify.
$$16 - 3x$$
Problem 3: Calculate $-2^4$.
Show Answer
Step 1: Understand the order of operations.
Indices (powers) happen before subtraction (the minus sign).
Step 2: Calculate the power first.
$$2^4 = 2 [[times]] 2 [[times]] 2 [[times]] 2 = 16$$
Step 3: Apply the minus sign.
Result: $-16$
Interactive Quiz
Can you dodge the minus sign traps?