1. Problem statement Problem: No problem was provided by the user, so I will solve an example quadratic equation to illustrate the method. 2. Formula and rules For quadratic equations $ax^2+bx+c=0$ use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Important rules: compute the discriminant $\Delta = b^2 - 4ac$ and consider cases $\Delta>0$ (two distinct real roots), $\Delta=0$ (one real repeated root), and $\Delta<0$ (two complex roots). 3. Example problem Solve $x^2 - 5x + 6 = 0$. 4. Intermediate work Identify coefficients: $a=1$, $b=-5$, $c=6$. Compute discriminant: $\Delta = (-5)^2 - 4\cdot1\cdot6 = 25 - 24 = 1$. Apply the quadratic formula: $$x = \frac{-(-5) \pm \sqrt{1}}{2\cdot1} = \frac{5 \pm 1}{2}$$ Therefore $x = \frac{5+1}{2} = 3$ or $x = \frac{5-1}{2} = 2$. 5. Explanation in plain language We computed the discriminant to determine the number and type of roots. Since $\Delta=1>0$ there are two distinct real roots. We substituted the coefficients into the quadratic formula and simplified step by step. 6. Final answer The solutions are $x=3$ and $x=2$. If you want me to solve a specific problem, please paste it exactly as written and I will solve it step by step.