If $a, b, c\in R$ and equations $ax^2+bx+c=0$ & $x^2+3x+4$ have a common root then $\displaystyle \frac {a+c}{b}=\frac {4}{3}$.

For the two quadratic equations to have a common root, the condition is c 1 a 2 minus c 2 a 1 times x squared plus b 1 c 2 minus b 2 c 1 times x plus a 1 b 2 minus a 2 b 1 equals 0 must have real roots, giving the relation c squared minus b c plus a squared minus 4 a equals 0. Letting c divided by b equal k, this rearranges to b squared k squared minus b c minus 4 a plus a squared equals 0, which requires the discriminant b squared minus 16 b squared k squared plus 16 b squared k to be greater than or equal to zero, proving a plus c divided by b is less than or equal to 4 divided by 3. Therefore, the assertion stating a plus c divided by b equals 4 divided by 3 is incorrect. The reason is the valid condition for a common root to exist. Thus, the assertion is incorrect but the reason is correct.

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