Question

If the roots of the equation ax2 + 2cx + b = 0 are real and distinct, then show that the roots
of the equation x2- 2(a + b)x + a2 + b2 + 2c2 = 0 are non-real complex numbers.​

Answer 1

If the roots of x2−2cx+ab=0 are real and unequal

then discriminant D>0

⇒(−2c)2−4ab>0

⇒4c2−4ab>0

⇒c2>ab

now in quadratic equation

x2−2(a+b)x+a2+b2+2c2=0

discriminant D={−2(a+b)}2−4(a2+b2+2c2)

                         =4(a+b)2−4(a2+b2+2c2)

                         =4(2ab−2c2)            

                         =8(ab−c2) < 0  

Since discriminant is negative

∴ The roots of the given equation will be imaginary=0