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2007 SMT Algebra Round Problem 1

Revision as of 23:29, 28 October 2025 by Yuhao2012 (talk | contribs) (Created page with "==Problem== Find all real roots of <math>f</math> if <math>f\left(x^{\frac19}\right)=x^2-3x-4</math>. ==Solution== After factoring, we get <math>f\left(x^{\frac19}\right)=(x-...")
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Problem

Find all real roots of $f$ if $f\left(x^{\frac19}\right)=x^2-3x-4$.

Solution

After factoring, we get $f\left(x^{\frac19}\right)=(x-1)(x+4)$, so to make $(x-1)(x+4)=0$, $x=4$ or $-1$, so, we have $4^{\frac19}$ or $(-1)^{\frac19}$, and because $4^{\frac19}$ can't be simplified and $(-1)^{\frac19}=-1$, our answer is $x=\boxed{\mathrm{4^{\frac19} \text{or} -1}}$

~Yuhao2012

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