Irrational Roots of Quadratic Equations with Integer Coefficients

Since coefficients $a, b, c$ and $a, m, n$ are integers, irrational roots occur in conjugate pairs.

For $ax^2 + bx + c = 0$, roots are $3 \pm 2\sqrt{2}$.

• Sum of roots $= 6 = -b/a \implies b = -6a$.
• Product of roots $= (3)^2 - (2\sqrt{2})^2 = 9 - 8 = 1 = c/a \implies c = a$.

For $ay^2 + my + n = 0$, roots are $4 \pm 2\sqrt{3}$.

• Sum of roots $= 8 = -m/a \implies m = -8a$.
• Product of roots $= (4)^2 - (2\sqrt{3})^2 = 16 - 12 = 4 = n/a \implies n = 4a$.

Now calculate $\frac{b}{m} + \frac{c-2b}{n}$: $\frac{b}{m} = \frac{-6a}{-8a} = \frac{3}{4}$ $c - 2b = a - 2(-6a) = 13a$ $\frac{c-2b}{n} = \frac{13a}{4a} = \frac{13}{4}$ $\frac{b}{m} + \frac{c-2b}{n} = \frac{3}{4} + \frac{13}{4} = \frac{16}{4} = 4$