Fixing 5 Common Algebra Mistakes
Stop losing easy marks by fixing these 5 simple algebra errors.
🎯 Why this matters: Algebra is like a language. If you mispronounce a word, the meaning changes completely! These 5 common mistakes trip up almost everyone. Master them now, and you will breeze through your exams.
1. The “Fruit Salad” Error (Unlike Terms)
This is the most frequent mistake beginners make. Remember: you can only add or subtract terms if they have the exact same variable parts.
❌ The Mistake:
$$2a + 3b = 5ab$$
✅ The Correction:
Think of $a$ as Apples and $b$ as Bananas. If you have 2 Apples and 3 Bananas, you do not have 5 Apple-Bananas. You just have a bowl of fruit.
Correct Answer: $2a + 3b$ (It cannot be simplified further).
Another variation: Thinking $x$ and $x^2$ are the same.
$$x + x^2 \neq 2x^2$$
These are unlike terms. One is a line ($x$), the other is a square ($x^2$). They stay separate.
2. The “Invisible Negative” (Distribution)
When you see a negative number outside a bracket, it is like a “virus”—it must infect everything inside.
The Rule: $-a(b + c) = -ab – ac$
❌ The Mistake:
$$-2(x + 4) = -2x + 4$$
(Student forgot to multiply the 4 by the negative).
✅ The Correction:
Multiply $-2$ by $x$: $-2x$
Multiply $-2$ by $+4$: $-8$
Correct Answer: $-2x – 8$
3. The “Freshman’s Dream” (Squaring Brackets)
This mistake is so famous it has a name! Students often try to distribute an exponent like it’s a multiplier. It doesn’t work that way.
❌ The Mistake:
$$(x + 3)^2 = x^2 + 9$$
✅ The Correction:
Squaring means multiplying something by itself. You must write it out and expand (FOIL).
$$(x + 3)(x + 3)$$
First: $x [[times]] x = x^2$
Outer: $x [[times]] 3 = 3x$
Inner: $3 [[times]] x = 3x$
Last: $3 [[times]] 3 = 9$
Combine the middle terms: $3x + 3x = 6x$.
Correct Answer: $x^2 + 6x + 9$
4. Illegal Cancellation (Fractions)
You can only cancel things that are multiplied (factors), not things that are added (terms).
❌ The Mistake:
$$[[frac{x + 6}{2}]] = x + 3$$
(Wait! This one is actually correct, but usually done for the wrong reason. Let’s look at a real error:)
❌ The Real Mistake:
$$[[frac{x + 5}{x}]] = 1 + 5 = 6$$
(Student crossed out the $x$ on top and bottom).
✅ The Correction:
Imagine $x = 10$.
$$[[frac{10 + 5}{10}]] = [[frac{15}{10}]] = 1.5$$
If you cancel the $x$’s, you get $6$. $1.5$ is definitely not $6$!
Rule: If there is a plus or minus sign holding terms together, you cannot just pick one out to cancel.
5. Solving vs. Simplifying
Finally, know the difference between an Expression and an Equation.
- Expression ($2x + 4$): No equals sign. You can only simplify it. You cannot find $x$.
- Equation ($2x + 4 = 10$): Has an equals sign. You can solve for $x$.
The Mistake: Taking an expression like $2x + 4$ and randomly writing $= 0$ just to find an answer. Don’t invent equals signs!
Practice Problems
Test your skills by correcting these common errors.
Problem 1: Expand and simplify correctly: $-3(2x - 5)$.
Show Answer
Step 1: Identify the multiplier.
The multiplier is $-3$ (don't forget the negative!).
Step 2: Multiply the first term.
$$-3 [[times]] 2x = -6x$$
Step 3: Multiply the second term.
The second term is $-5$.
$$-3 [[times]] -5 = +15$$
Final Answer:
$$-6x + 15$$
Problem 2: Expand $(y - 4)^2$.
Show Answer
Step 1: Write it out fully.
$$(y - 4)(y - 4)$$
Step 2: Use FOIL (First, Outer, Inner, Last).
First: $y [[times]] y = y^2$
Outer: $y [[times]] -4 = -4y$
Inner: $-4 [[times]] y = -4y$
Last: $-4 [[times]] -4 = +16$
Step 3: Combine like terms.
$$-4y - 4y = -8y$$
Final Answer:
$$y^2 - 8y + 16$$
Interactive Quiz
Can you spot the correct algebra?