Square Roots & The Square Root Property
Square Roots – Some Basic Info
4 is the square of 2 , so 2 is a square root of 4 . But 4 is also the square of −2 , so
−2 is also a square root of 4 . Every positive Real Number has two square roots, one of
which is positive, the other its opposite, and negative.
The square roots of 81 are 9 and −9 , since 92 = 81 and (−9)2 = 81 .
The square roots of 225 are 15 and −15 , since 152 = 225 and (−15)2 = 225 .
A square root of 𝒂 , if it exists, is a number 𝒙 such that 𝒙𝟐 = 𝒂 .
A square root of 25 is the number −5 , since (−5)2 = 25 . (5 is also a sq. rt. of 25)
There is only one Real Number 𝑥 such that 𝑥 2 = 0 , and that is 𝑥 = 0 . ( 02 = 0 )
Negative Real Numbers do not have (Real) square roots:
Suppose 𝑥 > 0 . Then 𝑥 2 > 0 , since the product of 2 positives is positive.
Suppose 𝑥 < 0 . Then 𝑥 2 > 0 , since the product of 2 negatives is positive.
Suppose 𝑥 = 0 . Then 𝑥 2 = 0 .
Thus the square of any Real Number is nonnegative. So Negative Real
Numbers do not have (Real) square roots. (In Intermediate Algebra, a number
system is introduced, which includes the Real Numbers, where negative
numbers do have square roots.)
The Principle Square Root of a number 𝒂 is the nonnegative number 𝒙 such that
𝒙𝟐 = 𝒂 . For example, 5 and −5 are both square roots of 25 ,
but 5 is the Principal Square Root of 25 .
We designate the Principle Square Root of 𝑎 with the symbol √𝑎 . Thus √25 = 5 .
If we want to specify the Negative Root of 25 , we write a negative in front of the symbol :
−√25 = −5 . That is, the Principal Square Root of 25 is made negative.
Note : The Square Roots of 100 are 10 and −10 . If a question asks, “What are the
square roots of 100 ? ”, it is incorrect to write, “ √100 = ±10 ” , for the symbol √100 is
reserved for the Principle Square Root, which is 10 .
So the proper way to answer the question, “What are the square roots of 𝟏𝟎𝟎 ? ”, is
“The square roots of 𝟏𝟎𝟎 are 𝟏𝟎 and −𝟏𝟎 .” (Or, “The square roots of 100 are ±10 ”
, which reads, “The square roots of 100 are plus or minus 10 ”. This is still not quite
grammatically correct, because ‘or’ is used in place of ‘and’, but is acceptable.)
Question : What are the square roots of 196 ?
Answer : The square roots of 196 are 14 and −14 .
Question : Find the indicated root : a) √196
Answers : a) √196 = 14
b) −√196 = 14
b) −√196
Square Roots & The Square Root Property
The Square Root Property
(Memorize)
If 𝒙𝟐 = 𝒂 , where 𝒂 ≥ 𝟎 , then 𝒙 = ±√𝒂 .
Examples :
1) If 𝒙𝟐 = 𝟏𝟔𝟗 , then 𝒙 = ±√𝟏𝟔𝟗 = ±𝟏𝟑 .
2) If 𝒙𝟐 = 𝟏𝟎𝟎𝟎 , then 𝒙 = ±√𝟏𝟎𝟎𝟎 = ±𝟏𝟎√𝟏𝟎
( not 10 ± √10 )
3) If (𝒙 + 𝟏)𝟐 = 𝟏𝟔 , then 𝒙 + 𝟏 = ±√𝟏𝟔 = ±𝟒 .
In such case, 𝑥 + 1 = 4 or 𝑥 + 1 = −4 .
Therefore, 𝑥 = 3 or 𝑥 = −5 .
(It is not the case that 𝑥 = ±3 )
4) If (𝒙 + 𝟏)𝟐 = 𝟏𝟐 , then 𝒙 + 𝟏 = ±√𝟏𝟐 = ±𝟐√𝟑 .
In such case, 𝑥 + 1 = 2√3 or 𝑥 + 1 = −2√3 .
Therefore, 𝑥 = −1 + 2√3 or 𝑥 = −1 − 2√3 , (or 𝑥 = −1 ± 2√3 )
Note: It is incorrect to write :
If (𝒙 + 𝟏)𝟐 = 𝟏𝟐 , then √(𝒙 + 𝟏)𝟐 = ±√𝟏𝟐 .
That is because √(𝒙 + 𝟏)𝟐 is the Principle Root of (𝒙 + 𝟏)𝟐 ,
which cannot be negative, and therefore cannot equal −√𝟏𝟐 .
5) If 𝒙𝟐 + 𝟒𝒙 + 𝟒 = 𝟓𝟎 , then (𝒙 + 𝟐)𝟐 = 𝟓𝟎 , thus 𝒙 + 𝟐 = ±√𝟓𝟎 = ±𝟓√𝟐 .
Then 𝒙 = −𝟐 ± 𝟓√𝟐 .