The quotient rule is typically used in calculus for differentiating functions, not for solving quadratic equations. However, you might be referring to solving quadratic equations by the method of factoring or by completing the square. I'll provide examples based on that assumption. If you meant something else, please clarify.

Example 1: Solving by Factoring

Equation:
$x^2 - 5x + 6 = 0$

Solution:

1. Factor the quadratic: $(x - 2)(x - 3) = 0$
2. Set each factor equal to zero:
   • $x - 2 = 0 \Rightarrow x = 2$
   • $x - 3 = 0 \Rightarrow x = 3$

Final Answer:
$x = 2$ or $x = 3$

Example 2: Solving by Completing the Square

Equation:
$x^2 + 6x + 5 = 0$

Solution:

1. Move the constant term to the right side:
   $x^2 + 6x = -5$
2. Add $\left(\frac{6}{2}\right)^2 = 9$ to both sides:
   $x^2 + 6x + 9 = 4$
3. Factor the left side:
   $(x + 3)^2 = 4$
4. Take the square root of both sides:
   $x + 3 = \pm 2$
5. Solve for $x$:
   • $x + 3 = 2 \Rightarrow x = -1$
   • $x + 3 = -2 \Rightarrow x = -5$

Final Answer:
$x = -1$ or $x = -5$

Example 3: Solving by Factoring

Equation:
$2x^2 - 8x = 0$

Solution:

1. Factor out the common term:
   $2x(x - 4) = 0$
2. Set each factor equal to zero:
   • $2x = 0 \Rightarrow x = 0$
   • $x - 4 = 0 \Rightarrow x = 4$

Final Answer:
$x = 0$ or $x = 4$

Example 4: Solving by Completing the Square

Equation:
$x^2 - 4x + 1 = 0$

Solution:

1. Move the constant term to the right side:
   $x^2 - 4x = -1$
2. Add $\left(\frac{-4}{2}\right)^2 = 4$ to both sides:
   $x^2 - 4x + 4 = 3$
3. Factor the left side:
   $(x - 2)^2 = 3$
4. Take the square root of both sides:
   $x - 2 = \pm\sqrt{3}$
5. Solve for $x$:
   • $x - 2 = \sqrt{3} \Rightarrow x = 2 + \sqrt{3}$
   • $x - 2 = -\sqrt{3} \Rightarrow x = 2 - \sqrt{3}$

Final Answer:
$x = 2 + \sqrt{3}$ or $x = 2 - \sqrt{3}$

Example 5: Solving by Factoring

Equation:
$3x^2 - 12x + 9 = 0$

Solution:

1. Factor the quadratic:
   $3(x^2 - 4x + 3) = 0$
2. Factor the expression inside the parentheses:
   $3(x - 3)(x - 1) = 0$
3. Set each factor equal to zero:
   • $x - 3 = 0 \Rightarrow x = 3$
   • $x - 1 = 0 \Rightarrow x = 1$

Final Answer:
$x = 3$ or $x = 1$

---

Would you like more details on any of these methods, or do you have any other questions?

Here are 5 related questions for further practice:

1. Solve the quadratic equation $x^2 - 9x + 20 = 0$ by factoring.
2. Complete the square to solve $x^2 + 10x + 21 = 0$.
3. Solve $4x^2 - 16x = 0$ by factoring.
4. Find the roots of $x^2 - 6x + 8 = 0$ using the quadratic formula.
5. Solve the equation $x^2 + 2x + 1 = 0$ by completing the square.

Tip: When completing the square, always remember to add and subtract the same value inside the equation to maintain equality.