https://file.aiquickdraw.com/mathbot/file/1727022936031nthbz10y.jpg

The image shows a quadratic equation problem where you are asked to solve by completing the square.

The given equation is:

\[
x^2 + 14x + 98 = 0
\]

The steps provided involve completing the square. Here's how you can solve it:

### Step-by-Step Solution:

1. **Rewrite the original equation** as:
   
   \[
   x^2 + 14x = -98
   \]
   This step is essentially isolating the \(x^2 + 14x\) terms on one side of the equation.

2. **Complete the square:**
   
   To complete the square, take half of the coefficient of \(x\) (which is 14), square it, and add it to both sides. Half of 14 is 7, and \(7^2 = 49\).
   
   \[
   x^2 + 14x + 49 = -98 + 49
   \]
   So, the equation becomes:
   
   \[
   (x + 7)^2 = -49
   \]

3. **Solve the equation:**
   
   Taking the square root of both sides:
   
   \[
   x + 7 = \pm \sqrt{-49}
   \]
   \[
   x + 7 = \pm 7i
   \]
   (since \(\sqrt{-49} = 7i\)).

4. **Final answer:**
   
   Solving for \(x\):
   
   \[
   x = -7 \pm 7i
   \]

### Filling in the blanks:
- Rewrite the equation: \( x^2 + 14x = -98 \)
- Add the appropriate number: \( x^2 + 14x + 49 = -98 + 49 \)
- Factor the left side: \( (x + 7)^2 = -49 \)
- Final answer: \( x = -7 \pm 7i \)

Would you like any further details or have any questions?

Here are 5 related questions for further exploration:
1. What is the process for completing the square on any quadratic equation?
2. How do imaginary numbers like \(i\) arise in quadratic equations?
3. How can we use the quadratic formula to check the solution of this equation?
4. What are the differences between completing the square and factoring quadratics?
5. In what scenarios is completing the square more useful than other methods?

**Tip:** Completing the square is especially useful for solving quadratic equations that do not factor easily and for deriving the quadratic formula.

Fill in all blanks when solving the following equation by completing the square: x^2 + 14x + 98 = 0.

In this problem, solve the quadratic equation x^2 + 14x + 98 = 0 by completing the square. Learn the step-by-step process of converting the equation into a perfect square trinomial and solving for x using imaginary numbers. Ideal for students in Grades 9-11.

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