Example 3.   Solve for x:

	(x + 3)²	=	5
Solution.    This has the form
	z²	=	a.
Therefore we solve it in the same manner as Example 2.
	x + 3	=	±
	x	=	−3 ±

Problem 6.   Solve for x.

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a)	(x − 1)²	=	2		b)	(x + 5)²	=	6
	x − 1	=	±			x + 5	=	±
	x	=	1 ±			x	=	−5 ±

c)	(x − p)²	=	q + r
	x − p	=	±
	x	=	p ±

The definition of the square root radical

Here is the formal rule that implicitly defines the symbol :

A square root radical multiplied by itself
produces the radicand.

Examples.	·	=	2
	(3a)²	=	3²a²()²   (Power of a product of factors)
		=	9a²· 10
		=	90a².

Problem 7.   Evaluate the following.

a)	·   =  3		b)	·   =  5
c)	()²  =  a + b		d)	=
e)	(5)²  =  25· 2 = 50		f)	(a4)²  =  a8· 3b = 3a8b

Example 4.   Multiply out  ( + ).  That is, distribute .

Solution.	( + )	=	·  + ·
		=	2 + 3

Problem 8.   Following the previous Example, multiply out

(  + ).

(  + )	=
	=

Rationalizing a denominator

Rationalizing a denominator is a simple technique for changing an irrational denominator into a rational one.  We simply multiply the radical by itself -- but then we must multiply the numerator by the same number.

Example 5.   Rationalize this denominator:

1

Solution.  Multiply both the numerator and denominator by :

The denominator is now rational.

2	can also take the form ½:

2

= ½

Real numbers

A real number is any number that you would expect to find on the number line.  The absolute value of each real number names a distance from 0.

The two main categories of real numbers are rational and irrational.  These numbers are the subject of calculus and of scientific measurement.

A real variable is a variable that takes on real values.

Problem 12.   Let x be a real variable, and let 3 < x < 4.  Name five values that x might have.

For example, 3.1,  3.14,  ,  ,  .

Problem 13.   If the square root is to be a real number, then the radicand may not be negative.  (There is no such real number, for example, as .)

If is to be real, then we must have  x ≥ 0.

(If you are not viewing this page with Internet Explorer 6, then your browser may not be able to display the symbol ≥, "is greater than or equal to;" or ≤, "is less than or equal to.")

Therefore, what values are permitted to the real variable x ?

a)      x − 3 ≥ 0; that is, x ≥ 3.

b)       1 + x ≥ 0;  x ≥ −1.