1. Let's solve an example quadratic equation: $x^2 - 5x + 6 = 0$. 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$. 3. We can solve it by factoring, using the quadratic formula, or completing the square. Here, factoring is straightforward. 4. Factor the quadratic: $x^2 - 5x + 6 = (x - 2)(x - 3) = 0$. 5. Set each factor equal to zero: $x - 2 = 0$ or $x - 3 = 0$. 6. Solve for $x$: $x = 2$ or $x = 3$. 7. Therefore, the solutions are $x = 2$ and $x = 3$. --- 1. Next, solve a linear equation: $3x + 4 = 19$. 2. Subtract 4 from both sides: $3x = 15$. 3. Divide both sides by 3: $x = 5$. 4. The solution is $x = 5$. --- 1. Finally, solve a system of equations: $$\begin{cases} 2x + y = 7 \\ x - y = 1 \end{cases}$$ 2. Add the two equations to eliminate $y$: $$ (2x + y) + (x - y) = 7 + 1 \Rightarrow 3x = 8 $$ 3. Solve for $x$: $x = \frac{8}{3}$. 4. Substitute $x$ back into $x - y = 1$: $$ \frac{8}{3} - y = 1 \Rightarrow y = \frac{8}{3} - 1 = \frac{5}{3} $$ 5. The solution is $x = \frac{8}{3}$, $y = \frac{5}{3}$.