If a and b are distinct real numbers, show that the quadratic equation

1.	If a and b are distinct real numbers, show that the quadratic equation 2(a2 + b2) x2 + 2 (a + b) x + 1 = 0 has no real roots.
Answer» 2(a² + b²) x² + 2 (a + b) x + 1 = 0 Compare given equation with the general form of quadratic equation, which is ax² + bx + c = 0 a = 2 (a² + b²), b = 2(a + b), c = 1 Discriminant: D = b² – 4ac =[2(a + b)]² – 4. 2 (a² + b²).1 = 4a² + 4b² + 8ab – 8a² – 8b² = – 4a² – 4b² + 8ab = – 4(a² + b² – 2ab) = – 4(a – b)² < 0 Hence the equation has no real roots.

If a and b are distinct real numbers, show that the quadratic equation 2(a2 + b2) x2 + 2 (a + b) x + 1 = 0 has no real roots.

Answer»

2(a² + b²) x² + 2 (a + b) x + 1 = 0

Compare given equation with the general form of quadratic equation, which is ax² + bx + c = 0

a = 2 (a² + b²), b = 2(a + b), c = 1

Discriminant:

D = b² – 4ac

=[2(a + b)]² – 4. 2 (a² + b²).1

= 4a² + 4b² + 8ab – 8a² – 8b²

= – 4a² – 4b² + 8ab

= – 4(a² + b² – 2ab)

= – 4(a – b)² < 0

Hence the equation has no real roots.