Question

How many quadratic polynomials with real coefficients are there such that the set of roots equals the set of coefficients? (For clarification: If the polynomial is $ax^2+bx+c,\ a\neq0,$ and the roots are $r$ and $s,$ then the requirement is that $\{a,b,c\}=\{r,s\}.$ )

$3$

$4$

$5$

$6$

infinitely many

Solution:

The set $\{a,b,c\}$ must equal the two-element set $\{r,s\},$ so at least two coefficients coincide, and the roots are the two distinct coefficient values. By Vieta's formulas $r+s=-\dfrac{b}{a}$ and $rs=\dfrac{c}{a}.$

Working through the cases of which coefficients are equal yields the polynomials $x^2+x-2,$ $-x^2-x,$ $x^2-\dfrac12 x-\dfrac12,$ and $ux^2+\dfrac1u x+u$ where $u$ is the unique real root of $u^3+u+1=0.$

That is $4$ polynomials in all.

Thus, B is the correct answer.

2019 AMC 12B Problem 21

Solution:

Problem 21 in Other Years