Common Student’s Mistakes in Mathematics
I NDICES
This tells your teacher that you are bad at math.
(𝑥 + 5)2 𝐼𝑆 𝑁𝑂𝑇 𝑥 2 + 25
Proof:
(𝑥 + 5)2 = (𝑥 + 5)(𝑥 + 5)
= (𝑥 + 5)(𝑥 + 5)
= 𝑥 2 + 10𝑥 + 25
Another example:
(𝑥 + 𝑦)2 = 𝑥 2 + 𝑦 2
Proof:
= (𝑥 + 𝑦)(𝑥 + 𝑦)
= 𝑥 2 + 2𝑥𝑦 + 𝑦 2
𝑥 2 + 2𝑥𝑦 + 𝑦 2 𝐼𝑆 𝑁𝑂𝑇 𝑥 2 + 𝑦 2
You can multiply them here, because inside the parenthesis the variables are
multiplied by each other
(𝑥 2 𝑦 3 )2 = 𝑥 4 𝑦 6
Proof:
(𝑥 2 𝑦 3 )2 = (𝑥 2 𝑦 3 )(𝑥 2 𝑦 3 )
= 𝑥2 ∗ 𝑥2 ∗ 𝑦3 ∗ 𝑦3
= 𝑥 2+2 𝑦 3+3
Therefor:
= 𝑥 4𝑦6
The same thing applies to square roots,
2
√𝑥 2 + 𝑦 2 = 𝑥 + 𝑦?
2
𝑜𝑟 √𝑥 2 + 25 = 𝑥 + 5?
You cannot simplify those equations
2
𝑡𝑎𝑘𝑒 √32 + 32
2
= √9 + 9
2
= √18
≈ 4.24
2
𝑛𝑜𝑤 𝑡𝑟𝑦 √32 + 32
2
= √3 2 + 3 2
=3+3
=6
4.24 ≠ 6
2
√3 2 + 3 2 ≠ 3 + 3
What about this?
2
2
2
√𝑥 + 𝑦 ≠ √ 𝑥 + √𝑦
Try to prove it by changing [x, y] to integers.
2
√3 + 6
2
= √9
=3
2
2
√3 + √6
= 1.73 + 2.45 = 4.18
3 ≠ 4.18
2
2
√3 + 6 ≠ √3 + √6
2
F RACTIONS
Fractions is where most students don’t really understand. or worse invent
new mathematics
2
3
3
4
𝑖𝑠 𝑖𝑡
2 3
3
÷
𝑜𝑟 2 ÷ 3 ÷ 𝑜𝑟 … ?
3 4
4
First of all, look at the biggest fraction sign
2
3
3
4
=
2 3
÷
3 4
Now if you don’ remember elementary math, to divide two fractions. Flip the
sign to multiplication (*) then flip the 2nd fraction. Finally, multiply.
2 3
÷
3 4
2 4
= ∗
3 3
8
=
9
You can make sure by inputting the operation into calculator like this
2 ÷ 3 = 0. 6̇
3 ÷ 4 = 0.75
0. 6̇ ÷ 0.75 = 0. 8̇
8 ÷ 9 = 0. 8̇
Therefor
2
3
3
4
=
2 3
÷
3 4
Try the other form of fractions and see the answer, it’ll be different thus
implying which is the correct one.
Canceling in fractions, could you cross out any number/variable in this
equation? Well, let’s try
3𝑥 + 4
𝑥+4
3𝑥 + 4
=3
𝑥+4
Now let’s do it again using the same fraction but different form
→
3𝑥 + 4
𝑥+4
=
3𝑥
4
+
𝑥+4 𝑥+4
It is the same fraction, but we cannot cancel them out? So how come we did
in first solution?
3𝑥
4
+
≠3
𝑥+4 𝑥+4
This implies that in fractions, addition and subtraction not inside
parenthesis [for example (x+5)] cannot be crossed out or cancelled.
2𝑥(𝑦 + 𝑧)
𝑥(𝑦 − 𝑧)
We can cancel out here
2𝑥(𝑦 + 𝑧) 2(𝑦 + 𝑧)
=
(𝑦 − 𝑧)
𝑥(𝑦 − 𝑧)
Because:
(𝑦 + 𝑧)
2𝑥(𝑦 + 𝑧)
2𝑥
=
∗
(𝑦 − 𝑧)
𝑥(𝑦 − 𝑧)
𝑥
𝐿𝑒𝑎𝑑𝑖𝑛𝑔 𝑡𝑜
2(𝑦 + 𝑧)
(𝑦 − 𝑧)
𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟
2𝑥(𝑦 + 𝑧)
𝑥(𝑦 − 𝑧)
𝑐𝑎𝑛 𝑏𝑒 𝑐𝑎𝑛𝑐𝑒𝑙𝑙𝑒𝑑 𝑜𝑢𝑡
However, don’t forget that if this equation becomes the following
You cannot cancel out x here. With the same reasoning above
2𝑥(𝑦 + 𝑧)
2
≠
𝑥(𝑦 + 𝑧) + 1
1
But you can cross out in this equation
2𝑥(𝑦 + 𝑧)
𝑥(𝑦 + 𝑧)
Notice how inside the parenthesis are the same in fraction? You can cancel
them out
2𝑥(𝑦 + 𝑧) 2𝑥
=
=2
𝑥(𝑦 + 𝑧)
𝑥
F ACTORIZING
Let’s take this equation as an example, what’s the first step?
𝑥 2 − 5𝑥 = 6𝑥
1) We bring all terms to the left side to make the right side equal 0
𝑥 2 − 5𝑥 = 6
𝑥 2 − 5𝑥 − 6 = 0
What about this?
𝑥 2 − 2𝑥 − 5 = 10
As stated above in step 1. Avoid putting terms to different side which will
make you prone to more silly mistakes
This is a longer and more thoughtless way to solve it:
𝑥 2 − 2𝑥 − 5 = 10
𝑥 2 − 2𝑥 = 10 + 5
𝑥 2 − 2𝑥 = 15
How would you solve this one?
𝑥2 = 𝑥
This is a guaranteed way to lose your marks.
𝑥2 𝑥
=
𝑥
𝑥
𝑥2 𝑥
𝑥2
=
=
𝑥
𝑥
𝑥
𝑇ℎ𝑖𝑠 𝑖𝑠 𝑤𝑟𝑜𝑛𝑔
Most of students make mistakes in those questions when it’s easier compared
to other questions. To solve this equation properly, as stated in step 1
previously. Move all same terms to one side
𝑥2 = 𝑥
𝑥2 − 𝑥 = 0
Great, now we can factorize
𝑥2 − 𝑥 = 0
𝑥(𝑥 − 1) = 0
𝑥 = 0 𝑜𝑟 𝑥 = 1
Common basic arithmetic mistake given this fraction and calculating it using a
calculator
6
5𝜋
Input in calculator: 6 ÷ 5𝜋 is wrong. Following ‘BODMAS’. The 6 divided by 5 will
be computed first.
Correct input into the Calculator is
6
(5𝜋)
We can say the same thing for 𝑥 2 = 𝑦
Put -17 for x. but instead put (−17)2 so the calculator computes -17 squared with
the sign.