Question

If the roots of the equation `ax^2+2bx+c=0` are real and distinct then find the nature of the roots of the equation `(a+c)(ax^2+2bx+c) = 2(ac - b²)(x²+1)`?

Answer 1

The roots are complex conjugate

Explanation:

If the roots of

`a x^2 + 2 b x + c=0`

are real and distint then `b^2-ac>0`

Now grouping

`(a + c) (a x^2 + 2 b x + c) - 2 (a c - b^2) (x^2 + 1)=0`

we have

`(a^2 + 2 b^2 - a c)x^2+2 b (a + c)x+2 b^2 + c (c-a) = 0`

and solving for `x`

#x = (-b (a + c) + sqrt[(4 b^2 + (a - c)^2) (a c-b^2)])/( a^2 + 2 b^2 - a c)#

and `a c-b^2<0` so the roots are complex conjugate