Question

If ad is not equal to bc, then prove that the equation (a^2 + b^2 )x^2 + 2(ac + bd) x + (c^2 + d^2 ) = 0 has no real roots.​

Answer 1

Step-by-step explanation:

A = (a²+b²),B =2(ab+cd) and C=(c²+d²)

D = B²-4AC

=2(ac+bd)²-4(a²+b²)(c²+d²)

=(4(a²c²+2abcd +b²d²))- 4(a²c²+a²d²+b²c²+b²d²)

=4a²c²+8abcd+4b²d²-4a²c²- 4a²d-4b²c²-4b²d²

=8abcd -4a²d²-4b²c²

=-4(4a²d²+4b²c²-2abcd)

=-4(ad-bc)²

hence the equation has no real roots.