Exercise 946Algebra › Equations and radicalsL1Added 3 Nov 2026

Vieta's formulas

#Vieta#sum and product of roots#quadratic

Statement

Let $x_1,x_2$ be the roots of $x^2-5x+3=0$ . Compute $x_1^2+x_2^2$ and $x_1^3+x_2^3$ using Vieta's formulas.

Hint

$x_1+x_2=5$ , $x_1x_2=3$ . Use $x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2$ .

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Vieta. $S=x_1+x_2=5$ , $P=x_1x_2=3$ .

$x_1^2+x_2^2$ . $=S^2-2P=25-6=19$ .

$x_1^3+x_2^3$ . Use $x^3+y^3=(x+y)^3-3xy(x+y)=S^3-3PS=125-45=80$ .

Conclusion. $x_1^2+x_2^2=19$ and $x_1^3+x_2^3=80$ .