Let equations $\displaystyle ax^{2}+bx+c=0 \left ( a,b,c \in R \right ) $ & $\displaystyle x^{2}+2x+5=0$ have a common root, then $\displaystyle \frac{a+c}{b}= \frac{1}{3}$.

The roots of the quadratic equation x^2 + 2x + 5 = 0 are found using the discriminant, yielding complex roots -1 + 2i and -1 - 2i. For the real quadratic equation ax^2 + bx + c = 0 to have a common root with this equation, it must share both complex roots, meaning the two equations are identical up to a constant multiple. Thus, a = k, b = 2k, and c = 5k for some non-zero real number k. Calculating the given expression (a + c) / b results in (k + 5k) / 2k = 6k / 2k = 3, not 1/3. Therefore, the assertion is incorrect, while the reason regarding complex roots appearing in conjugate pairs for real polynomials is correct.

• Thanks
• Wrong explanation