Quadratic Equations
Roots Transformation — New Quadratic from Derived Roots
nta_pyq_2024_apr
Grade 11
Question:
If 2 and 6 are the roots of the equation $ax^2+bx+1=0$, then the quadratic equation, whose roots are $\dfrac{1}{2a+b}$ and $\dfrac{1}{6a+b}$, is:
$2x^2+11x+12=0$
$x^2+8x+12=0$
$4x^2+14x+12=0$
$x^2+10x+16=0$
Step-by-Step Solution
Key Concept: Sum of roots $=8=-b/a$, product $=12=1/a$, so $a=1/12$, $b=-2/3$. Compute $2a+b=1/6-2/3=-1/2$ and $6a+b=1/2-2/3=-1/6$.
$a=1/12$, $b=-2/3$. New roots $-2$ and $-6$. $x^2+8x+12=0$.
Correct Answer: 2