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Roots equals Coefficients

Source: 2019 AMC 12B #21

February 14, 2019
2019 AMC 12BAMCAMC 12AMC 12 Bquadraticsalgebrapolynomial

Problem Statement

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 ax2+bx+c,a0,ax^2+bx+c,a\neq 0, and the roots are rr and s,s, then the requirement is that {a,b,c}={r,s}\{a,b,c\}=\{r,s\}.)
<spanclass=latexbold>(A)</span>3<spanclass=latexbold>(B)</span>4<spanclass=latexbold>(C)</span>5<spanclass=latexbold>(D)</span>6<spanclass=latexbold>(E)</span>infinitely many<span class='latex-bold'>(A) </span> 3 \qquad<span class='latex-bold'>(B) </span> 4 \qquad<span class='latex-bold'>(C) </span> 5 \qquad<span class='latex-bold'>(D) </span> 6 \qquad<span class='latex-bold'>(E) </span> \text{infinitely many}