1. The problem is to solve a quadratic equation without using the algebraic formula (quadratic formula). 2. Instead, we can use factoring, completing the square, or graphing methods depending on the equation. 3. For example, consider the quadratic equation $x^2 - 5x + 6 = 0$. 4. Step 1: Try factoring the quadratic. We look for two numbers that multiply to $6$ and add to $-5$. 5. These numbers are $-2$ and $-3$ because $-2 \times -3 = 6$ and $-2 + -3 = -5$. 6. Step 2: Factor the quadratic as $(x - 2)(x - 3) = 0$. 7. Step 3: Set each factor equal to zero: $x - 2 = 0$ or $x - 3 = 0$. 8. Step 4: Solve each equation: $x = 2$ or $x = 3$. 9. Therefore, the solutions to the quadratic equation are $x = 2$ and $x = 3$. 10. This method avoids using the quadratic formula and relies on factoring.