Solve the quadratic equation that involves sum and product

[chwala]

Homework Statement

see attached

Relevant Equations

quadratic equations

I am refreshing on this...Have to read broadly...i will start with (b) then i may be interested in alternative approach or any correction that may arise from my working. Cheers.

Kindly note that i do not have the solutions to the following questions...

For (b), we know that,
say, if $x=α$ and $x=β$ are roots of the given quadratic equation, then it follows that,
$-(α+β)=3.5$
$αβ=2$,
$(α-β)^2=(α+β)^2-4αβ$
$(α-β)^2=(3.5)^2-8$
$(α-β)^2=\dfrac{17}{4}$

For (c),
$α^3+β^3=(α+β)(α^2-αβ+β^2)$
$α^2+β^2=(α+β)^2-2αβ$
$α^2+β^2=(3.5)^2-4$
$α^2+β^2=\dfrac{33}{4}$
Therefore,
$α^3+β^3=(α+β)(α^2+β^2-αβ)$
$α^3+β^3=(-3.5)(8.25-2)$
$α^3+β^3=(-3.5)(6.25)$
$α^3+β^3=-\dfrac{175}{8}$

For part (d),
$α^3-β^3=\sqrt{\dfrac {17}{4}}⋅\dfrac {41}{4}=\dfrac {41}{4}⋅{\dfrac {\sqrt17}{2}}=\dfrac {41}{8}\sqrt 17$

For part (e),
We shall have,
$x^2-\dfrac{α^3+β^3}{αβ}x+αβ=0$
$x^2-\dfrac{175}{16}x+2=0$
$⇒16x^2-175x+32=0$

Bingo!

 

[Mark44]

Parts b and c agree with the results I got, and pretty much follow the approach I used.