Posted by Jayden Haddy on Sunday, August 11, 2013 at 12:52am.
In quadratic equation:
a x ^ 2 + b x + c
expresion :
delta = b ^ 2 - 4 a c
is called the Discriminant
When delta > 0
There is two real root
When delta = 0
There is 2 equal real roots. ( The root is repeated. )
When delta < 0
There are two complex ( unreal ) roots.
In this case :
a = m ^ 2 + n ^ 2
b = - 2 ( m + n )
c = 2
delta = b ^ 2 - 4 ac
delta = [ - 2 ( m + n ) ] ^ 2 - 4 * ( m ^ 2 + n ^ 2 ) * 2
delta = 4 * ( m + n ) ^ 2 - 8 * ( m ^ 2 + n ^ 2 )
delta = 4 * ( m ^ 2 + 2 m n + n ^ 2 ) - 8 m ^ 2 - 8 n ^ 2
delta = 4 m ^ 2 + 8 m n + 4 n ^ 2 - 8 m ^ 2 - 8 n ^ 2
delta = - 4 m ^ 2 + 8 m n - 4 n ^ 2
delta = - 4 ( m ^ 2 - 2 m n + n )
delta = - 4 ( m - n ) ^ 2
Becouse : ( m - n ) ^ 2 = m ^ 2 - 2 * m n + n ^ 2
So delta = - 4 ( m - n ) ^ 2
When m > n
( m - n ) are positive
Square of positive number are positive number.
- 4 * positive number are negative number
When m < n
( m - n ) are negative
Square of negative number are positive number.
- 4 * positive number are negative number
This mean when m is not equal to n the Disciminant is always negative , and your equation always have two unreal roots.
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Remark :
delta is greek letter delta ( Like triangle).
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