1.

If a and b are real and a ≠ b then show that the roots of the equation (a - b)x2 + 5(a + b)x - 2(a - b) = 0. are equal and unequal.

Answer»

the given equation is (a - b)x2 + 5(a + b)x - 2(a - b) = 0

∴ D = [5(a + b)]2 - 4 x (a - b) x [-2(a - b)]

= 25(a + b)2 + 8(a - b)2

Since a and b are real and a ≠ b, so (a - b)2 > 0 and (a + b)2 > 0

∴8(a - b)2 > 0 ......... (1) (Product of two positive numbers is always positive)

Also, 25(a + b)2 > 0 .......(2) (Product of two positive numbers is always positive)

Adding (1) and (2), we get

25(a + b)2 + 8(a - b)2 > 0 (Sum of two positive numbers is always positive)

⇒ D > 0

Hence, the roots of the given equation are real and unequal.



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