Difference between revisions of "2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2"
(→Solution) |
(→Solution) |
||
Line 12: | Line 12: | ||
Finally, <math>f(x) = (x-2)^3 - 2 = \sqrt[9]{2} \Rightarrow x = \sqrt[3]{2+\sqrt[9]{2}}</math>. | Finally, <math>f(x) = (x-2)^3 - 2 = \sqrt[9]{2} \Rightarrow x = \sqrt[3]{2+\sqrt[9]{2}}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | Note that <math>f(x)=(x+2)^3-2</math>. Thus, <math>f^{n}(x)=(x+2)^{3n}-2</math>. In the question, <math>n=3</math>, so we have <math>(x+2)^{27}-2=0</math>. Solving, <math>\boxed{x=-2+\sqrt[27]{2}}</math> | ||
+ | |||
+ | ~yofro | ||
== See also == | == See also == |
Revision as of 23:12, 3 August 2020
Contents
Problem
If , solve the equation
Solution
We can write . This means that .
Working backwards, .
Then, .
Finally, .
Solution 2
Note that . Thus, . In the question, , so we have . Solving,
~yofro
See also
2014 UNM-PNM Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNM-PNM Problems and Solutions |