Relative powers
The square of a number is 12 more than the number itself. The cube of the number is 9 times the number. What is the number?
Age
14 to 16
Challenge level
Problem
The square of a number is 12 more than the number itself.
The cube of the number is 9 times the number.
What is the number?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $-3$
Let the number be $y$.
Then $y^3 = 9y\\
y^2\times y = 9\times y$, so either $y=0$ or $y^2 = 9$, giving $y = 3$ or $y = -3$.
$y^2$ is $12$ more than $y$
$y=0\Rightarrow y^2=0$
$y=3$ or $y=-3$ means $y^2=9$ which is $12$ more than $-3$