Solve the given quadratic equation that involves sum and product

[chwala]

Homework Statement

See attached below;

Relevant Equations

sum/product

For part a,
We have $α+β=b$ and $αβ =c$. It follows that,
$(α^2 + 1)(β^2+1)=α^2β^2+α^2+β^2+1)$
=$α^2β^2+(α+β)^2-2αβ +1$
=$c^2+b^2-2c+1$
=$c^2-2c+1+b^2$
=$(c-1)^2+b^2$

For part b,..we shall have
$x^2- \dfrac{α+β+α^2β +αβ^2}{(α^2 + 1)(β^2+1)}$ $x$ +$\dfrac {αβ}{(α^2 + 1)(β^2+1)}$
$x^2-\dfrac{α+β+αβ(β +α)}{(α^2 + 1)(β^2+1)}$$x$+$\dfrac {αβ}{(α^2 + 1)(β^2+1)}$
$x^2-\dfrac{b(1+c)}{(c-1)^2+b^2}$$x$+$\dfrac {c}{(c-1)^2+b^2}$

Is this correct?( i do not have the solutions)...i would appreciate different ways of attempting the problem. Cheers.

 

[anuttarasammyak]

It seems all right.