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Vieta's Formula

Vieta's formula relates the coefficients of polynomials to the sum and products of their roots. This can provide a shortcut to finding solutions in more complicated algebraic polynomials.

Level 4

         

If the roots of \( x^{2} + qx + p = 0 \) are \( \sqrt{6 + 2\sqrt{5}}\) and \(\sqrt{6 - 2\sqrt{5}},\) what is \(p - q?\)

Consider the quadratic equation \[{ ax }^{ 2 }+bx+c=0\] with roots \(\alpha\) and \(\beta\), and whose coefficients \(a,b,c\) are distinct, non-zero real numbers in arithmetic progression.

If \(\frac { 1 }{ \alpha } +\frac { 1 }{ \beta } ,\ \alpha +\beta, \ { \alpha }^{ 2 }+{ \beta }^{ 2 }\) form a geometric progression, find \(\frac { a }{ c }.\)

Let \(f(x) = x^3 - ax^2 + bx - b\) for some positive integers \(a\) and \(b.\) If the roots of \(f(x)=0\) are distinct positive integers, what is the value of \(a + b?\)

Let \(x, y, z\) be the real roots of the cubic equation

\[2u^3-799u^2-400u-1=0\]

and let \(\omega = \tan^{-1} x+\tan^{-1} y+\tan^{-1} z\). If \(\tan \omega = \frac{a}{b}\), where \(a\) and \(b\) are positive coprime integers, what is the value of \(a+b\)?

This problem is posed by Russelle G.

Consider the equation \[x^4 - 18x^3 +kx^2 +174x -2015.\] If the product of two of the four roots of the equation is \(-31\), then find the value of \(k\).

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