AIEEE 2008 (Offline)Question 1
The quadratic equations $x^{2}-6x+a=0$ and $x^{2}-cx+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4:3$. Then the common root is
Quadratic Equation · Common Roots
Solution
Concept. Let the common root be $\alpha$ and the other roots $4k$ and $3k$; use the known sum of the first and product of the second. Formulas used. (1) Sum and product of rootsStep 1 - Equations. First: $\alpha+4k=6$. Second: $\alpha\cdot3k=6$, i.e. $\alpha k=2$. Step 2 - Solve. $(6-4k)k=2$ gives $2k^2-3k+1=0$, so $k=1$ or $\frac12$. Step 3 - Integer roots. $k=\frac12$ gives $2$ and $\frac32$, not both integers. So $k=1$ and $\alpha=6-4=2$. Answer: (D)