Show that if the roots of the following quadratic equation are equal, then ad = bc

Question

Philoid · Accepted Answer

Compare the given quadratic equation with ax² + bx + c = 0

Here,

The coefficient of x² is a

The coefficient of xis b

And, c is the intercept.

Now, here:

a = a² + b²

b = 2(ac + bd)

c = c² + d²

Consider,

Then,

⇒ b² – 4ac = [2(ac + bd)]² – 4(a² + b²)(c² +d²)

= 4[a²c² + 2abcd + b²d²] – 4[a²c² + a²d² + b²c² + b²d²]

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

= 8abcd– 4a²d² – 4b²c²

= –4[4a²d² + 4b²c² – 2abcd]

= –4[ad – bc]²

Hence the given equation has no real roots unless ad = bc