Given that \( f(x) \) is a polynomial of degree 2, let \( f(x) = ax^2 + bx + c \) where \( a \neq 0 \).
Using the given condition:
\( f(x)f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right) \)
Substituting the polynomial expression:
\( (ax^2 + bx + c)\left(a\frac{1}{x^2} + b\frac{1}{x} + c \right) = (ax^2 + bx + c) + \left( a\frac{1}{x^2} + b\frac{1}{x} + c \right) \)
After simplifying, the resulting equation becomes:
\( 1 - K^2 = -2K \)
The equation simplifies to:
\( K^2 - 2K - 1 = 0 \)
Solving this quadratic equation:
\( K = \frac{2 \pm \sqrt{(2)^2 - 4(1)(-1)}}{2(1)} \)
\( K = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2} \)
Using the identity:
\( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \)
From Vieta’s formulas:
\( \alpha + \beta = 2 \quad \text{and} \quad \alpha\beta = -1 \)
Thus,
\( \alpha^2 + \beta^2 = 2^2 - 2(-1) = 4 + 2 = 6 \)
If \[ \int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left( \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| \right) + C, \] where \( C \) is the constant of integration, then \( \alpha + 2\beta \) is equal to ________________
If the system of equations \[ x + 2y - 3z = 2, \quad 2x + \lambda y + 5z = 5, \quad 14x + 3y + \mu z = 33 \]
has infinitely many solutions, then \( \lambda + \mu \) is equal to:
If the equation of the parabola with vertex \( \left( \frac{3}{2}, 3 \right) \) and the directrix \( x + 2y = 0 \) is \[ ax^2 + b y^2 - cxy - 30x - 60y + 225 = 0, \text{ then } \alpha + \beta + \gamma \text{ is equal to:} \]
The steam volatile compounds among the following are: