Algebra

Vieta's Formula

Vieta's Formula: Level 4 Challenges

         

If the roots of x2+qx+p=0 x^{2} + qx + p = 0 are 6+25 \sqrt{6 + 2\sqrt{5}} and 625,\sqrt{6 - 2\sqrt{5}}, what is pq?p - q?

Consider the quadratic equation ax2+bx+c=0{ ax }^{ 2 }+bx+c=0 with roots α\alpha and β\beta, and whose coefficients a,b,ca,b,c are distinct, non-zero real numbers in arithmetic progression.

If 1α+1β, α+β, α2+β2\frac { 1 }{ \alpha } +\frac { 1 }{ \beta } ,\ \alpha +\beta, \ { \alpha }^{ 2 }+{ \beta }^{ 2 } form a geometric progression, find ac.\frac { a }{ c }.

Let f(x)=x3ax2+bxbf(x) = x^3 - ax^2 + bx - b for some positive integers aa and b.b. If the roots of f(x)=0f(x)=0 are distinct positive integers, what is the value of a+b?a + b?

Let x,y,zx, y, z be the real roots of the cubic equation

2u3799u2400u1=02u^3-799u^2-400u-1=0

and let ω=tan1x+tan1y+tan1z\omega = \tan^{-1} x+\tan^{-1} y+\tan^{-1} z. If tanω=ab\tan \omega = \frac{a}{b}, where aa and bb are positive coprime integers, what is the value of a+ba+b?

This problem is posed by Russelle G.

Consider the equation x418x3+kx2+174x2015.x^4 - 18x^3 +kx^2 +174x -2015. If the product of two of the four roots of the equation is 31-31, then find the value of kk.

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