AIEEE 2003 (Offline)Question 1
The value of $a$ for which one root of the quadratic equation $\left(a^{2}-5a+3\right)x^{2}+(3a-1)x+2=0$ is twice as large as the other is
Quadratic Equation · Relation Between Roots and Coefficients
Solution
Concept. With roots $r$ and $2r$, eliminate $r$ between the sum and the product. Formulas used. (1) Sum and product of rootsStep 1 - Equations. Let $A=a^2-5a+3$. Then $3r=-\dfrac{3a-1}{A}$ and $2r^2=\dfrac2A$, i.e. $r^2=\dfrac1A$. Step 2 - Eliminate $r$. $9r^2=\dfrac9A=\dfrac{(3a-1)^2}{A^2}$, so $9A=(3a-1)^2$. Step 3 - Solve. $9a^2-45a+27=9a^2-6a+1$, so $39a=26$ and $a=\frac23$. Answer: (B)