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# Problems using Vieta's formulas: Difficult Problems with Solutions

Problem 1
If $$x_1, x_2$$ are the roots of the equation $$x^2+5x-3=0$$, determine the value of $$x_1^2+x_2^2$$.
Problem 2
If $$x_1, x_2$$ are the roots of the equation $$x^2+11x+12=0$$, determine the value of $$x_1^2+x_2^2$$.
Problem 3
If $$x_1, x_2$$ are the roots of the equation $$x^2+9x+33=0$$, determine the value of $$\frac{1}{x_1}+\frac{1}{x_2}$$.
Problem 4
If $$x_1, x_2$$ are the roots of the equation $$x^2-8x+11=0$$, determine the value of $$x_1^3+x_1^2+x_1+x_2^3+x_2^2+x_2$$.
Problem 5
If $$x_1, x_2$$ are the roots of the equation $$x^2-15x+36=0$$, determine the value of $$|x_1-x_2|$$.
Problem 6
Let $$x_1, x_2$$ be the roots of the equation $$x^2-12x+19=0$$. Determine the value of $$x_1(1-x_1)+x_2(1-x_2)$$.
Problem 7
If $$x_1, x_2$$ are the roots of the equation $$x^2-4x+1=0$$, determine the value of $$(x_1-\frac{1}{x_1})^2+(x_2-\frac{1}{x_2})^2$$.
Problem 8
If $$x_1, x_2$$ are the solutions to the equation $$x^2-5x+a^2-2a+1=0$$ where $$a \in R$$. Find the value of a, for which $$x_1x_2$$ is minimal.
Problem 9 sent by Berenguer Sabadell
Find the value of $\alpha_1^2+\alpha_2^2+\alpha_3^2$ where $\alpha_1$, $\alpha_2$ and $\alpha_3$ are the roots of the equation $3x^3-2x^2+5x-7=0$.
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