Question

If the roots of the equation x²+ 2cx+ ab=0 are real unequal ,prove that the equation
x²-2(a+b )x + a² + b² + 2c² =0 has no real roots

Answer 1

Step-by-step explanation:

if x² + 2cx + ab=0 have real unequal roots then b²-4ac>0

i.e. (2c)²-4(1)(ab)>0

4c²-4ab>0

4c²>4ab

c²>ab                   ......................( 1 )

in x²-2(a+b)x+a²+b²+2c²=0

b²-4ac = (-2(a+b))² - 4(1)(a²+b²+2c²)

           = 4(a²+b²+2ab)-4(a²+b²+2c²)

           =4a²+4b²+8ab-4a²-4b²-8c²

           =8ab-8c²

           =8(ab-c²)

but since ab<c² then ab-c²<0

therefore, x²-2(a+b)x+a²+b²+2c²=0 has no real roots as  b²-4ac<0