Quadratic Equations
Roots of derived equation from symmetric functions
MJMT_Full_Test_01
Grade 12
Question:
Let $\alpha,\beta$ be roots of $x^2-3x-\sqrt7=0$ and $\alpha^2+\frac1\beta,\beta^2+\frac1\alpha$ be roots of $x^2+px+q=0$. For $x^2+p^2qx-(11q+p)=0$, roots are
positive
negative
one positive and other negative
none of these
Step-by-Step Solution
Key Concept: $\alpha+\beta=3$, $\alpha\beta=-\sqrt7$. New roots sum $=\alpha^2+\beta^2+1/\alpha+1/\beta=(\alpha+\beta)^2-2\alpha\beta+(\alpha+\beta)/\alpha\beta=9+2\sqrt7-3/\sqrt7=-p$. Product $=q$. Compute $p$ and $q$, then analyze roots of final equation.
One positive, one negative.
Correct Answer: 3