Finding the quadratic equation from its given roots. 
If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c
=0$ , then form an equation whose roots are:
$\alpha+\dfrac{1}{\beta},\beta+\dfrac{1}{\alpha}$

Now, using Vieta's formula, 
For new equation, 
Product of roots ($P$) = $\dfrac{a^2+c^2+2ac}{ac}$
Sum of roots ($S$) = $\dfrac{2c-b}{c}$
Hence, 
the required equation is:
$acx^2 - ax(2c-b)+(a+c)^2=0$                       // as quadratic equation = $x^2-Sx +Px=0$
But the answer key states that the answer is: 
$acx^2 +b(a+c)x
+(a+c)^2=0$ 
I am really doubtful of this answer. Where have I gone wrong (or is the answer in the key wrong?)?

$$S=\alpha+\beta+\frac{1}{\alpha}+\frac{1}{\beta}=(\alpha+\beta)\left(1+\frac{1}{\alpha\beta}\right)$$
If I write this in terms of $a,b,c$ I get $$S=-\frac{b}{a}\left(1+\frac{a}{c}\right)=-\frac{b(a+c)}{ac}$$
