2007 SMT Algebra Round Problem 1

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 roots $=\boxed{\mathrm{4^{\frac19} \text{ or } -1}}$ .

~Yuhao2012

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2007 SMT Algebra Round Problem 1

Problem

Solution