Quadratic Root Dependence
Find the number of quadratic equations of the form $ x^2 + ax + b = 0, $ such that whenever $ c $ is a root of the equation, $ c^2 - 2 $ is also a root of the equation.
- 1
- 2
- 3
- +
- 4
- 5
- 6
- -
- 7
- 8
- 9
- $\frac{a}{b}$
- .
- 0
- =
- %
- $a^n$
- $a^{\circ}$
- $a_n$
- $\sqrt{}$
- $\sqrt[n]{}$
- $\pi$
- $\ln{}$
- $\log$
- $\theta$
- $\sin{}$
- $\cos{}$
- $\tan{}$
- $($
- $)$
- $[$
- $]$
- $\cap$
- $\cup$
- $,$
- $\infty$
Solution
Let the roots be $ r $ and $ s $ (not necessarily real). We take the cases where $ r = s $ and $ r \neq s $.
Case 1: $ r = s $.
Since $ r $ is the only root, we must have $ r^2 - 2 = r $. Then $ r^2 - r - 2 = 0, $ which factors as $ (r - 2)(r + 1) = 0, $ so $ r = 2 $ or $ r = -1 $. This leads to the quadratics $ x^2 - 4x + 4 $ and $ x^2 + 2x + 1 $.
Case 2: $ r \neq s $.
Each of $ r^2 - 2 $ and $ s^2 - 2 $ must be equal to $ r $ or $ s $. We have three cases:
(i) $ r^2 - 2 = r $ and $ s^2 - 2 = s $.
(ii) $ r^2 - 2 = s $ and $ s^2 - 2 = r $.
(iii) $ r^2 - 2 = s^2 - 2 = r $.
In case (i), as seen from Case $ r, $ $ s \in \{2,-1\} $. This leads to the quadratic $ (x - 2)(x + 1) = x^2 - x - 2 $.
In case (ii), $ r^2 - 2 = s $ and $ s^2 - 2 = r $. Subtracting these equations, we get
\[r^2 - s^2 = s - r.\]Then $ (r - s)(r + s) = s - r $. Since $ r - s \neq 0, $ we can divide both sides by $ r - s, $ to get $ r + s = -1 $. Adding the equations $ r^2 - 2 = s $ and $ s^2 - 2 = r, $ we get
\[r^2 + s^2 - 4 = r + s = -1,\]so $ r^2 + s^2 = 3 $. Squaring the equation $ r + s = -1, $ we get $ r^2 + 2rs + s^2 = 1, $ so $ 2rs = -2, $ or $ rs = -1 $. Thus, $ r $ and $ s $ are the roots of $ x^2 + x - 1 $.
In case (iii), $ r^2 - 2 = s^2 - 2 = r $. Then $ r^2 - r - 2 = 0, $ so $ r = 2 $ or $ r = -1 $.
If $ r = 2, $ then $ s^2 = 4, $ so $ s = -2 $. (We are assuming that $ r \neq s $.) This leads to the quadratic $ (x - 2)(x + 2) = x^2 - 4 $.
If $ r = -1 $, then $ s^2 = 1, $ so $ s = 1 $. This leads to the quadratic $ (x + 1)(x - 1) = x^2 - 1 $.
Thus, there are $ \boxed{6} $ quadratic equations that work, namely $ x^2 - 4x + 4, $ $ x^2 + 2x + 1, $ $ x^2 - x - 2, $ $ x^2 + x - 1, $ $ x^2 - 4, $ and $ x^2 - 1 $.