Question

If p, q are real and $p\ne q,$ then show that the roots of the equation $(p-q)x^2+5(p+q)x-2(p-q)=0$ are real and unequal.

Answer 1

For an equation $a{x}^2+bx+c=0$ the nature of the roots can be determined by the value of D.
where $D=b^2-4ac$
If, D > 0, real and unequal roots

For the given equation $(p-q)x^2+5(p+q)x-2(p-q)=0$, roots are real and unequal.

Hence, its discriminant value must be greater than zero.

D= ${5(p+q)}^2-4(p-q)(-2)(p-q)$

= $25(p+q{)}^2+8(p-q{)}^2>0$ [ Sum of square of two numbers is always greater than zero]

Therefore, roots are real and unequal