jmc

Question

Compute the sum of the roots of the equation xx - 6x + 7x - 1 = 0,given that all of the roots are real and nonnegative.

Accepted Answer

The given equation is not a polynomial equation, so we can't use Vieta's formulas directly. To create a related polynomial equation, we substitute

y = x,

or

x = y^{2},

giving

y^{3} - 6 y^{2} + 7 y - 1 = 0.

For each value of

y

which satisfies this equation, the corresponding value of

x

which satisfies the original equation is

x = y^{2} .

Therefore, we want to find the sum of the squares of the roots of this equation.

To do this, let

r,

s,

and

t

denote the roots of this equation. Then by Vieta's formulas,

r + s + t = 6

and

r s + s t + t r = 7,

so

r^{2} + s^{2} + t^{2} = (r + s + t)^{2} - 2 (r s + s t + t r) = 6^{2} - 2 \cdot 7 = 22 .