×
Back to all chapters

# 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.

# Vieta's Formula: Level 4 Challenges

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 $$k$$.

×