Difference between revisions of "2007 SMT Algebra Round Problem 1"
(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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==Solution== | ==Solution== | ||
| − | After factoring, we get <math>f\left(x^{\frac19}\right)=(x-1)(x+4)</math>, so to make <math>(x-1)(x+4)=0</math>, <math>x=4</math> or <math>-1</math>, so, we have <math>4^{\frac19}</math> or <math>(-1)^{\frac19}</math>, and because <math>4^{\frac19}</math> can't be simplified and <math>(-1)^{\frac19}=-1</math>, our answer is <math> | + | After factoring, we get <math>f\left(x^{\frac19}\right)=(x-1)(x+4)</math>, so to make <math>(x-1)(x+4)=0</math>, <math>x=4</math> or <math>-1</math>, so, we have <math>4^{\frac19}</math> or <math>(-1)^{\frac19}</math>, and because <math>4^{\frac19}</math> can't be simplified and <math>(-1)^{\frac19}=-1</math>, our answer is roots <math>=\boxed{\mathrm{4^{\frac19} \text{ or } -1}}</math>. |
~Yuhao2012 | ~Yuhao2012 | ||
Latest revision as of 08:37, 29 October 2025
Problem
Find all real roots of 
if 
.
Solution
After factoring, we get 
, so to make 
, 
or 
, so, we have 
or 
, and because can't be simplified and 
, our answer is roots 
.
~Yuhao2012
