Question

If the roots of the equation x^2 + 2cx+ ab = 0 are real and equal then prove that x ^2 - 2 in (a + b) x + a ^2+ b^2 + 2c^2 is equal to zero has no real roots

Answer 1

for two distinct real roots, the discriminant > 0

ie for x^2 + 2cx + ab

4c^2 - 4ab > 0

c^2 - ab > 0

for unreal roots disc. < 0

ie for x^2 – 2(a + b)x + a^2 + b^2 + 2c^2

4(a + b)^2 - 4(a^2 + b^2 + 2c^2) < 0

(a + b)^2 - (a^2 + b^2 + 2c^2) < 0

a^2 + b^2 + 2ab - (a^2 + b^2 + 2c^2) < 0

2ab - 2c^2 < 0

ab - c^2 < 0

and we see this is now true