Algebra Mistakes

Here's a collection of algebra mistakes that are pretty easy to make.

Try to avoid them!

Mistake	Correction
x² = 25, so x = 5	x = 5 or x = −5
(x−5)² = x² − 25	= (x−5)(x−5) = x² − 10x + 25
√(x²+y²) = x + y	√(x²+y²) is as far as we can go

x²x⁴= x⁸	= x⁶ (add exponents)
(x²)⁴= x⁶	= x⁸ (multiply exponents)
2x^-1 = 1/(2x)	= 2/x
−5² = 25	= −25 (do exponent before minus)
(−5)² = −25	= +25 (do brackets before exponent)
5^½ = 1/5²	= √5

log(a+b) = log(a) + log(b)	log(a+b) is as far as we can go

x(a/b) = xa/xb	= xa/b
x−(5+a) = x−5+a	= x−5−a

And be careful of these ones too:

Simplifying Fractions

xx+y = xx + xy

We can't simplify that!

Imagine x=4 and y=5:

44+5 = 49

That's definitely not equal to 44 + 45 (which is more than 1)

Maybe you were thinking of this kind of fraction that we can simplify:

x+yx = xx + yx

Square root of xy

√(xy) =√x√y ... but not always!

Example: x = −5 and y = −2

√10 = √(−5 × −2) =√(−5)√(−2) (the mistake) =i√5 × i√2 =i²√5√2 =−√10

So, does √10 = −√10 ??? I think not!

√(xy) =√x√y only when x and y are both >= 0

Two Equals One

Example:

Start with: x = y Multiply both sides by x: x² = xy Subtract y² from both sides: x² − y² = xy − y² Factor:(x+y)(x−y) = y(x−y) Divide both sides by (x−y):x + y = y (the mistake) Since x = y, we see that: 2y = y And so: 2 = 1

Hang on! That can't be right!

What went wrong? Silly us! We tried to divide by zero.

When we said that x=y, it means that (x−y)=0 , so going from (x+y)(x−y) = y(x−y) to x + y = y is a mistake.

Factoring

Example: Solve x² – 5x = 2

Start with:x² – 5x = 2 Factor x:x(x−5) = 2 So:x=2 or x−5=2 (the mistake) And so:x=2 or 7

Let's check x=2:

2² – 5×2 = 4−10 = −6, but we wanted x² – 5x = 2

That only works when x(x−5) = 0 (zero) not any other number.

See Zero Product Property<>