prove that if m is not equal to n, the equation (m^2+n^2)x^2 - 2(m+n)x + 2=0 has unreal roots.

Answer 1

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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