1. **State the problem:** We are given a quadratic equation with roots and asked to find the value of $m$. The equation is $$X^2 - 4X + 3 + \frac{1}{m} = 0$$ 2. **Rewrite the equation:** Combine the constant terms: $$X^2 - 4X + \left(3 + \frac{1}{m}\right) = 0$$ 3. **Recall the sum and product of roots formulas:** For a quadratic equation $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$: - Sum of roots: $$r_1 + r_2 = -\frac{b}{a}$$ - Product of roots: $$r_1 r_2 = \frac{c}{a}$$ Here, $a=1$, $b=-4$, and $c=3 + \frac{1}{m}$. 4. **Calculate sum and product of roots:** $$r_1 + r_2 = -\frac{-4}{1} = 4$$ $$r_1 r_2 = 3 + \frac{1}{m}$$ 5. **Use the discriminant condition for real roots:** The discriminant $\Delta$ must be non-negative: $$\Delta = b^2 - 4ac \geq 0$$ Calculate: $$\Delta = (-4)^2 - 4 \times 1 \times \left(3 + \frac{1}{m}\right) = 16 - 4\left(3 + \frac{1}{m}\right)$$ Simplify: $$16 - 12 - \frac{4}{m} = 4 - \frac{4}{m}$$ 6. **Set discriminant to zero for equal roots (if roots are equal):** $$4 - \frac{4}{m} = 0$$ Divide both sides by 4: $$\cancel{4} - \frac{\cancel{4}}{m} = 0 \Rightarrow 1 - \frac{1}{m} = 0$$ 7. **Solve for $m$:** $$1 = \frac{1}{m}$$ Multiply both sides by $m$: $$m = 1$$ **Final answer:** $$m = 1$$