Question

if ad is not equal to bc then prove that (a^2+b^2)x^2+2(ac+bd)x+(c^2+d^2)=0has no real roots

Answer 1

Answer:

Hello.  Hope this helps.

The discriminant of the quadratic is

[ 2(ac+bd) ]²  -  4 ( a² + b² ) ( c² + d² )

= 4a²c² + 4b²d² + 8abcd - 4a²c² - 4a²d² - 4b²c² - 4b²d²

= 8abcd - 4a²d² - 4b²c²

= -4 ( a²d² + b²c² - 2abcd )

= -4 ( ad - bc )²

If ad ≠ bc, then this number is negative.

Since the discriminant is negative, there are no real roots.