The quadratic equations x2−6x+a=0 and x2−cx+6=0 have one root in common. The other roots of the first and second equations are integers in the ratio 4:3. The common root is
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Official solution
Let the roots of equation x2−6x+a=0 be \[\alpha \]
and 4 \[\beta \] and that of the equation
x2−cx+6=0 be \[\alpha \] and \[3\beta .\] Then
\[\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha + 4\beta = 6;\,\,\,\,\,\,\,4\alpha \beta = a\]
and \[\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha + 3\beta = c;\,\,\,\,\,\,\,3\alpha \beta = 6\]
\[ \Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a = 8\]
\[\therefore\] The equation becomes
x2−6x+8=0
\[ \Rightarrow \left( {x - 2} \right)\left( {x - 4} \right) = 0\]
\[ \Rightarrow \] roots are 2 and 4
\[ \Rightarrow \alpha = 2,\beta = 1\]
\[\therefore\] Common root is 2.