# Math Help - Algebra 1

1. ## 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.

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

3. Originally Posted by xwrathbringerx
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 $|xy| = |x||y| \implies (ac-bd)^2 + (ad+bc)^2 = (a^2+b^2)(c^2+d^2)$.