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answer number 3 using complete the square

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.

Solve the quadratic equation x^2 + 5x - 14 = 0 using the completing the square method.

Learn how to solve the quadratic equation x^2 + 5x - 14 = 0 using the completing the square method. This step-by-step explanation covers key algebraic concepts such as factoring and properties of squares, suitable for Grades 8-10.

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