Download Square root?

√
Oh Hoon Kwon
Algebra Textbooks usually show the following shortened steps for solving a quadratic equation.
x2 = 36 √
(1)
x = (± 36) = ±6
This is a correct way of writing solutions on the blackboard. However, teachers usually add
more steps verbally (take the square roots) or in a written form such as (∗) in the middle to
help students understand.
2
√x = 36
√
(2)
x2 =
36
(∗)
x = ±6
The writing above is still mathematically correct. However, there has been a misreading/misleading problem in such writing. Students are used to interpret the above differently
and contrarily to the writer’s intention. That is because students cannot see invisible/hidden
lines√of equivalent√equations that include
yet. As a result, some students
√ further calculations
√
2
2
see x = x and 36 = ±6 although x = |x| and 36 = 6. The missing calculations are
here.
2
√x = 36
√
x2 = √36
(∗)
|x| = ( 36
√
(3)
36 if x ≥ 0.
√
x =
− 36 if x < 0.
√
x = ± 36
x = ±6
There are many different ways to ask a square root of a number. The verbal expression,“a
√
square root of” is not exactly the same as the symbol, “ ”.
What is a square root of 36? That can be either 6 or −6. You cannot say ±6. Just pick
one up.
What are the square roots of 36? They are 6 and −6. You can say ±6. One is a positive
number whose square is 36, i.e., 6 in 62 = 36. The other is a negative number whose square
is 36, i.e., −6 in (−6)2 = 36.
What is the square root of 36, then? It is 6. When we say the square root of a positive
number, it means the principal square root among two square roots of a positive number
which is the positive number whose square is the given number.
Therefore, “the square root of 36” is always
1
√
36, and it is 6.