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Quadratic
equation
x2
=
a,
a
>
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 |
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b) (x
-
5)2
= 49 |
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x2
-
42 =
0 |
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(x
-
5)2
-
72 =
0 |
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(x
-
4)
· (x
+
4) =
0 |
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[(x
-
5) -
7] ·
[(x
-
5) +
7]
=
0 |
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x
-
4 =
0
=> x1
= 4 |
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(x
-
5) -
7 =
0
=> x1
= 12 |
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x
+
4 =
0
=> x2
= -
4 |
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(x
-
5) +
7 =
0 => x2
= -2 |
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Square
root |
The
principal square root of a
nonnegative real number
a,
denoted |
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represents the
nonnegative real |
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number
x
whose square (the result of multiplying the number by itself) is
a,
that is |
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The number a
under the root sign is called the radicand.
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As the square root is defined only
when its radicand is a
nonnegative number therefore,
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for
example, the expression |
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makes sense only if
x
-
5
>
0, that is,
only if x
>
5. |
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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.
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That is, by the square
root of a,
where a is
a nonnegative number, we should mean, the positive square root of
a.
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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
). |
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As
a2
> 0 whether
a
> 0 or
a
< 0 we write
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Examples: |
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Properties
of square roots |
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Adding,
subtracting, multiplying and dividing square roots |
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Rationalizing
a denominator |
Rationalizing a denominator is a method for changing an irrational denominator into a rational one. |
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To
rationalize a denominator or numerator of the form |
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multiply both numerator |
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and denominator by a
conjugate, where |
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are conjugates of each other. |
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