Question

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

Collected in the board: Vieta's Formula

Steven Zheng posted 2 hours ago

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
(1)

and

αβ = -4
(2)

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 \text{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

Steven Zheng posted 2 hours ago

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