If $$x, y$$ are real numbers and equations $$x^2 - 12x + 35 = 0$$ and $$x^2 + ax + 105 = 0$$ have at least one common root, then the minimum possible value of $$y^2 + 4y - 5a$$ is ____

$$x^2 - 12x + 35 = 0$$ has roots 5 and 7. So 5 or 7 is also a root of $$x^2 + ax + 105 = 0$$.

• If root = 5: $$25 + 5a + 105 = 0 \Rightarrow a = -26$$.
• If root = 7: $$49 + 7a + 105 = 0 \Rightarrow a = -22$$.

$$y^2 + 4y - 5a$$ minimum over real $$y$$ is at $$y = -2$$ giving $$-4 - 5a$$.

• $$a = -26 \Rightarrow -4 + 130 = 126$$.
• $$a = -22 \Rightarrow -4 + 110 = 106$$.

Minimum value = 106.