Show that (a^2+b^2)(c^2+d^2)=(ac+bd)^2 + (ad-bc)^2

Answer 1

Foil Left side gives you:
= a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2
= (ac)^2 + (ad)^2 + (bc)^2 + (bd)^2

Foil Right side (ac+bd)^2
= ac x ac + ac x bd + ac x bd + bd x bc
= (ac)^2 + 1acbd + 1acbd + (bd)^2

Foil (ad-bc)^2
= ad x ad - bc x ad - bc x ad + bc x bc
= (ad)^2 - 1acbd - 1acbd + (bc)^2

Combine all terms on right side
left with (ac)^2 + (bd)^2 + (ad)^2 + (bc)^2