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

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Answer 1 · 2016-07-27T12:35:21

The roots are complex conjugate

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

If the roots of the equation ax^2+2bx+c=0ax2+2bx+c=0ax^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)(a+c)(ax^2+2bx+c) = 2(ac - b²)(x²+1)?