Sincea =0thisgives
(
x+b
2 a) 2
=b^2 − 4 ac
( 2 a)^2Hence
x+b
2 a=±√
b^2 − 4 ac
2 aso x=−b
2 a±√
b^2 − 4 ac
2 a=−b±√
b^2 − 4 ac
2 aAlso, 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, becausethis 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