Algebra

Vieta's Formula

Vieta's Formula: Level 3 Challenges

         

Jason and Thompson were solving the quadratic equation x2+bx+c=0x^{2} + bx + c = 0.

Jason wrote down the wrong value of b,b, and found the roots to be 6 and 1. Thompson wrote down the wrong value of c,c, and found the roots to be 4-4 and 1.-1.

What are the actual roots of the equation?

The non-zero roots of

ax2+bx+c=0ax^2+bx+c=0

are rr and s.s. What are the roots of

cx2+bx+a=0?cx^2+bx+a = 0?

The roots of the polynomial equation x3+2x2+3x+4=0 x^3 + 2 x^2 + 3 x + 4 = 0 are α,β, \alpha, \beta, and γ\gamma. What is the value of

α+βγ+β+γα+γ+αβ? \frac{ \alpha + \beta} { \gamma} + \frac{ \beta + \gamma} { \alpha } + \frac{ \gamma + \alpha } { \beta } ?

If α\alpha, β\beta, and γ\gamma are the roots of x3x1=0x^3-x-1=0, compute:

1α1+α+1β1+β+1γ1+γ\frac{1-\alpha}{1+\alpha}+\frac{1-\beta}{1+\beta}+\frac{1-\gamma}{1+\gamma}

(x7)(x3)(x+5)(x+1)=1680 \large (x-7)(x-3)(x+5)(x+1)=1680

Find the sum of all xx that satisfy the equation above.

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