Let's solve the quadratic equation from number 3, $x^2 + 5x - 14 = 0$, using the "completing the square" method.

Steps to Complete the Square:

1. Start with the equation: $x^2 + 5x - 14 = 0$

2. Move the constant term to the other side: $x^2 + 5x = 14$

3. Take half of the coefficient of $x$ (which is 5), square it, and add it to both sides:

   • Half of 5 is $\frac{5}{2}$, and its square is $\left(\frac{5}{2}\right)^2 = \frac{25}{4}$.

   So, add $\frac{25}{4}$ to both sides: $x^2 + 5x + \frac{25}{4} = 14 + \frac{25}{4}$

4. Simplify the right side: Convert 14 to a fraction with denominator 4: $14 = \frac{56}{4}$

   Now add: $\frac{56}{4} + \frac{25}{4} = \frac{81}{4}$

   So the equation becomes: $\left(x + \frac{5}{2}\right)^2 = \frac{81}{4}$

5. Take the square root of both sides: $x + \frac{5}{2} = \pm \frac{9}{2}$

6. Solve for $x$:

   • For the positive root: $x + \frac{5}{2} = \frac{9}{2} \implies x = \frac{9}{2} - \frac{5}{2} = \frac{4}{2} = 2$

   • For the negative root: $x + \frac{5}{2} = -\frac{9}{2} \implies x = -\frac{9}{2} - \frac{5}{2} = -\frac{14}{2} = -7$

Final Solutions:

$x = 2 \quad \text{or} \quad x = -7$

This completes the solution of the quadratic equation $x^2 + 5x - 14 = 0$ using the method of completing the square.

Would you like any more details or clarifications? Here are five follow-up questions:

1. Can you apply the quadratic formula to verify these solutions?
2. What is the geometric interpretation of completing the square?
3. How would the solutions differ if we had a different constant term on the right-hand side?
4. Can completing the square always be used to solve any quadratic equation?
5. How does completing the square relate to finding the vertex form of a quadratic equation?

Tip: Completing the square is useful for rewriting quadratics into a form that easily reveals the vertex of the parabola.