Sincea =0thisgives
(
x+
b
2 a
) 2
=
b^2 − 4 ac
( 2 a)^2
Hence
x+
b
2 a
=±
√
b^2 − 4 ac
2 a
so x=−
b
2 a
±
√
b^2 − 4 ac
2 a
=
−b±
√
b^2 − 4 ac
2 a
Also, once the square is completed for a quadratic it becomes clear what its maximum or
minimum values are asxvaries. This is becausexonly occurs under a square, which is
always positive. Looking at the general form:
a
[(
x+
b
2 a
) 2
−
b^2 − 4 ac
( 2 a)^2
]
we have two cases:
a positive.a> 0 /
Asxvaries, the quadratic will go through aminimumvalue whenx+
b
2 a
=0, because
this yields the smallest value within the square brackets.
a negative.a< 0 /
Asxvaries, the quadratic goes through amaximumvalue whenx+
b
2 a
=0.
Example
For the minimum value of 3x^2 + 5 x−2wehave
3 x^2 + 5 x− 2 ≡ 3
[(
x+^56
) 2
−
( 7
6
) 2 ]
This has a minimum value (3 is positive) whenx=−
5
6
. The minimum value is 3
(
−
(
7
6
) 2 )
=−
49
12
.
There is a useful relationship between the roots of a quadratic equation and its coeffi-
cients. Thus, supposeα,βare the roots of the quadraticx^2 +ax+b.Thenx−α,x−β