HMMT 2016 T7

Let $q(x)=q^{1}(x)=2 x^{2}+2 x-1$ , and let $q^{n}(x)=q\left(q^{n-1}(x)\right)$ for $n>1$ . How many negative real roots does $q^{2016}(x)$ have?

Dataset text NuminaMath olympiads. Quoted with source.

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\frac{2^{2017}+1}{3}

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