

Quadratic equation x^{2}
= a,
a
>
0 
Radicals (Roots) 
Square
root 
Properties
of square roots 
Adding,
subtracting, multiplying and dividing square roots 
Rationalizing
a denominator 





Quadratic
equation
x^{2}
=
a,
a
>
0 
The quadratic equation
x^{2}
=
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)
x^{2}
= 16 

b) (x

5)^{2}
= 49 

x^{2
}
4^{2 }=
0 

(x

5)^{2
}
7^{2 }=
0 

(x

4)
· (x
+
4) =
0 

[(x

5) 
7] ·
[(x

5) +
7]
=
0 

x

4 =
0
=> x_{1}
= 4 

(x

5) 
7 =
0
=> x_{1}
= 12 

x
+
4 =
0
=> x_{2}
= 
4 

(x

5) +
7 =
0 => x_{2}
= 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
a^{2
}
> 0 whether
a^{
}
> 0 or
a^{
}< 0 we write


Examples: 







Properties
of square roots 


Adding,
subtracting, multiplying and dividing square roots 


Rationalizing
a denominator 
Rationalizing a denominator is a method for changing an irrational denominator into a rational one. 

To
rationalize a denominator or numerator of the form 

multiply both numerator 

and denominator by a
conjugate, where 

are conjugates of each other. 









Intermediate
algebra contents 



Copyright
© 2004  2020, Nabla Ltd. All rights reserved. 