Normal

Problems using Vieta's formulas: Difficult Problems with Solutions

Problem 1
If [tex]x_1, x_2[/tex] are the roots of the equation [tex]x^2+5x-3=0[/tex], determine the value of [tex]x_1^2+x_2^2[/tex].
Problem 2
If [tex]x_1, x_2[/tex] are the roots of the equation [tex]x^2+11x+12=0[/tex], determine the value of [tex]x_1^2+x_2^2[/tex].
Problem 3
If [tex]x_1, x_2[/tex] are the roots of the equation [tex]x^2+9x+33=0[/tex], determine the value of [tex]\frac{1}{x_1}+\frac{1}{x_2}[/tex].
Problem 4
If [tex]x_1, x_2[/tex] are the roots of the equation [tex]x^2-8x+11=0[/tex], determine the value of [tex]x_1^3+x_1^2+x_1+x_2^3+x_2^2+x_2[/tex].
Problem 5
If [tex]x_1, x_2[/tex] are the roots of the equation [tex]x^2-15x+36=0[/tex], determine the value of [tex]|x_1-x_2|[/tex].
Problem 6
Let [tex]x_1, x_2[/tex] be the roots of the equation [tex]x^2-12x+19=0[/tex]. Determine the value of [tex]x_1(1-x_1)+x_2(1-x_2)[/tex].
Problem 7
If [tex]x_1, x_2[/tex] are the roots of the equation [tex]x^2-4x+1=0[/tex], determine the value of [tex](x_1-\frac{1}{x_1})^2+(x_2-\frac{1}{x_2})^2[/tex].
Problem 8
If [tex]x_1, x_2[/tex] are the solutions to the equation [tex]x^2-5x+a^2-2a+1=0[/tex] where [tex]a \in R[/tex]. Find the value of a, for which [tex]x_1x_2[/tex] 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 $3x^3-2x^2+5x-7=0$.
Normal
Submit a problem on this page.

Correct:
Wrong:
Unsolved problems:
Contact email:
Follow us on   Twitter   Facebook

Copyright © 2005 - 2020.