Algebra (Period 4)

RADICAL EXPRESSIONS 11-2
If an expression under the SQ RT sign is NEGATIVE, it does not exist in the REAL numbers!
There is no number that you can square and get a NEGATIVE PRODUCT

VARIABLES UNDER THE SQUARE ROOT SIGN:
If you have a variable under the SQ RT sign,
you need to determine what values of the variable will keep the radicand greater than or equal zero
The square root of x then is only real when x is greater than or equal to zero
The square root of (x + 2) is only real when x + 2 is greater than or equal to 0
Set x + 2 greater than or equal to 0 and solve as an inequality!
You will find that x must be greater than or equal to -2

SPECIAL CASE!!!! a variable squared plus a positive integer under radical:
If you're trying to find the principal square root of x2 (or any variable squared) plus a positive integer, then all numbers will work because a squared number will always end up either positive or zero!
Example: √(x ² + 3) under the radical, any number positive or negative will keep the radicand positive (real), because once you square it, it is positive.
Then you're just adding another positive number.

If there is a variable squared and then a negative number (subtraction), the square will need to be equal or greater than that negative number to stay zero or positive under the radical.
EXAMPLE: √(x² - 10)
x² must be equal or greater than 10, so x must be at least the square root of 10
(the square root of 10 squared is 10)

ANOTHER SPECIAL CASE!!!!!!!!!!
ANY RADICAL EXPRESSION THAT HAS A VARIABLE SQUARED IS SIMPLIFIED TO THE ABSOLUTE VALUE OF THE VARIABLE.
Example: The square root of x² is the absolute value (positive) of x ( lxl )
Why?
Because it is assumed that you're finding the PRINCIPAL (positive) square root.
EXAMPLE:
x = -3
√x² = √(-3)² = √9 = 3 (not -3)
so you have to put absolute value signs around the answer
IF THERE IS A VARIABLE SQUARED
(see p. 489 #17-30)

TRINOMIALS UNDER THE RADICAL:
What do you think you would do if you saw x² + 10x + 25 under the radical sign????
FACTOR IT!
IT MAY BE A PERFECT SQUARE (a binomial squared!)

EXAMPLE:
√( x² + 10x + 25) factors to √(x + 5)² = l x + 5l