Quadratic equation  x2 = aa > 0
     Radicals (Roots)
      Square root
         Properties of square roots
         Adding, subtracting, multiplying and dividing square roots
         Rationalizing a denominator
Quadratic equation  x2 = aa > 0
The quadratic equation x2 = a has two solutions which are opposite numbers only if a > 0.
If a = 0 then there is a repeated solution zero, and if  a < 0 then there are no solutions in the set of real numbers.
The solutions are also called the roots or zeros of the quadratic equation.
Examples: a)   x2 = 16   b)   (x - 5)2 = 49
        x2 - 42 = 0         (x - 5)2 - 72 = 0
       (x - 4) (x + 4) = 0         [(x - 5) - 7] [(x - 5) + 7] = 0
        x - 4 = 0    =>    x1 = 4         (x - 5) - 7 = 0    =>    x1 = 12
        x + 4 = 0    =>    x2 = - 4         (x - 5) + 7 = 0    =>    x2 = -2
Square root
The principal square root of a nonnegative real number a, denoted   represents the nonnegative real 
number x whose square (the result of multiplying the number by itself) is a, that is 
   
The number a under the root sign is called the radicand. 
As the square root is defined only when its radicand is a nonnegative number therefore, 
for example, the expression makes sense only if  x - 5 > 0, that is, only if x > 5.
Since the positive square root of a positive real number is called the principal square root this is why the square root of say 16, is taken as +4 despite the fact that the square of -4 is also 16
That is, by the square root of a, where a is a nonnegative number, we should mean, the positive square root of a.
The reason for taking the principal square root as the value of a square root of a positive real number, lies in the definition of a function f which requires that for each x in the domain there is at most one pair (x, y) in the codomain (where a set of ordered pairs (x, y) represents the graph of the function f ).
As a2 > 0 whether a > 0 or a < 0 we write
   
Examples:  
   
Properties of square roots
Properties Examples
       
       
       
         
         
         
         
Adding, subtracting, multiplying and dividing square roots
Examples:  
   
   
   
   
   
Rationalizing a denominator
Rationalizing a denominator is a method for changing an irrational denominator into a rational one.
Examples:  
   
   
To rationalize a denominator or numerator of the form multiply both numerator 
and denominator by a conjugate, where are conjugates of each other.
College algebra contents
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