Sum of Squares

Algebra Level 3

Find the sum of the squares of the three solutions of the equation $x^3 + 3x^2 - 7x + 1 = 0$ .

1 solution

Arron Kau Staff
May 13, 2014

Let $r_1$ , $r_2$ , and $r_3$ denote the three solutions. Using Vieta's formulas, we have $r_1+r_2+r_3 = -3$ and $r_1r_2 + r_2 r_3 + r_3 r_1 = -7$ .

Thus $(r_1+r_2+r_3)^2 = 9$ , and so $r_1^2 + r_2^2 + r_3^2 + 2(r_1r_2 + r_2r_3 + r_3r_1) = 9$ .

Therefore $r_1^2 + r_2^2 + r_3^2 + 2(-7) = 9$ and $r_1^2 + r_2^2 + r_3^2 = 23$ .

You haven't proved that all the roots are different. Common roots are counted as multiple but common solutions are taken as one.

mietantei conan - 7 years ago