1. **State the problem:** Solve the quadratic equation $$x^2-(k+1)x+4k-3=0$$ for $x$ in terms of the parameter $k$. 2. **Recall the quadratic formula:** For an equation $$ax^2+bx+c=0$$, the solutions are given by $$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ where $a=1$, $b=-(k+1)$, and $c=4k-3$ in this problem. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-(k+1))^2 - 4 \times 1 \times (4k-3) = (k+1)^2 - 4(4k-3)$$ 4. **Simplify the discriminant:** $$ (k+1)^2 - 16k + 12 = (k^2 + 2k + 1) - 16k + 12 = k^2 - 14k + 13 $$ 5. **Write the solutions using the quadratic formula:** $$x = \frac{(k+1) \pm \sqrt{k^2 - 14k + 13}}{2}$$ 6. **Interpretation:** The solutions depend on $k$ and are real if the discriminant $k^2 - 14k + 13 \geq 0$. **Final answer:** $$x = \frac{k+1 \pm \sqrt{k^2 - 14k + 13}}{2}$$