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proof that the equation
x^2(a^2+b^2)+2x(ac+bd)+(C^2+d^2)=0 has no real roots.

  • maths -

    the discriminant is

    4(ac+bd)^2 - 4(a^2+b^2)(c^2+d^2)
    junk the 4, as it does not change things, and you have

    a^2c^2 + 2abcd + b^2d^2 - a^2c^2 - a^2d^2 - b^2c^2 - b^2d^2
    = 2abcd - a^2d^2 - b^2c^2
    = -(ad-bc)^2

    since that is always negative, there are no real roots.

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