# 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)?

Jul 27, 2016

The roots are complex conjugate

#### Explanation:

If the roots of

$a {x}^{2} + 2 b x + c = 0$

are real and distint then ${b}^{2} - a c > 0$

Now grouping

$\left(a + c\right) \left(a {x}^{2} + 2 b x + c\right) - 2 \left(a c - {b}^{2}\right) \left({x}^{2} + 1\right) = 0$

we have

$\left({a}^{2} + 2 {b}^{2} - a c\right) {x}^{2} + 2 b \left(a + c\right) x + 2 {b}^{2} + c \left(c - a\right) = 0$

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