If $$\alpha ,\beta$$ are two roots of the equation $$x^2+2x-4=0$$, then find the value of $$α^3-3β^2+2β$$.

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Uniteasy Admin · Accepted Answer

Since $$\alpha ,\beta$$ are two roots of the equation $$x^2+2x-4=0$$, the following equations are obtained by using Vieta's formula.
$$α+β = -2 $$n
and
$$αβ = -4$$n
Substitute β to the equation, we get another equation
$$β^2+2β-4=0$$ and $$α^2+2α-4=0$$
or
$$β^2=4-2β \quad 	ext{or} \quad α^2 = 4-2α$$ 
which can be used for power reducing of square terms.
$$α^3-3β^2+2β$$
$$=αα^2-3β^2+2β$$
$$=α( 4-2α)-3(4-2β)+2β$$
$$=4α-2α^2-12+8β$$
$$=4α-2( 4-2α)-12+8β$$
$$=8(α+β)-20$$
$$ =-16-20$$
$$=-36$$

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