Question

If a and b are real and a ≠ b then show that the roots of the equation 

`(a-b)x^2+5(a+b)x-2(a-b)=0`are equal and unequal. 

Answer 1

The given equation is `(a-b)x^2+5(a+b)x-2(a-b)=0` 

∴ D=`[5(a+b)]^2-4xx(a-b)xx[-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.