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Math Help - Algebra 1

  1. #1
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    Exclamation Algebra 1

    Show that (a^2 + b^2)(c^2 + d^2) = (ac bd)^2 + (ad + bc)^2. Hence, or otherwise, write the number 17000 as the sum of two squares in two different ways.

    Can someone please help me on this? I've expanded the first equation but I can't seem to be able to factorise it into the other equation.
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  2. #2
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    Take Right side =(ac-bd)^2+(ad+bc)^2

    = a^2c^2-2abcd+b^2d^2+a^2d^2+2abcd+b^2c^2

    =a^2c^2+a^2d^2+b^2c^2+b^2d^2

    =a^2(c^2+d^2)+b^2(c^2+d^2)

    =(a^2+b^2)(c^2+d^2)

    =Left side
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  3. #3
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    Quote Originally Posted by xwrathbringerx View Post
    Show that (a^2 + b^2)(c^2 + d^2) = (ac bd)^2 + (ad + bc)^2. Hence, or otherwise, write the number 17000 as the sum of two squares in two different ways.
    Let x=a+bi and y=c+di.

    Then we have |xy| = |x||y| \implies (ac-bd)^2 + (ad+bc)^2 = (a^2+b^2)(c^2+d^2).
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