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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.

  • maths -

    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.

    ________________________________________

    Remark :

    delta is greek letter delta ( Like triangle).

    ________________________________________

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