How can I use completing the square to solve the equation x^2 + 12x + 49?

To solve the quadratic equation x² + 12x + 49 = 0 by completing the square, follow these steps:

1. First, rewrite the equation:

   x² + 12x + 49 = 0

2. Next, we want to isolate the constant term on one side to focus on completing the square. To do this, we can move the constant (49) to the right side:

   x² + 12x = -49

3. Now, we need to complete the square on the left side. Take the coefficient of x (which is 12), divide it by 2 (getting 6), and then square it (resulting in 36).

   So, we add 36 to both sides of the equation:

   x² + 12x + 36 = -49 + 36

4. This simplifies to:

   x² + 12x + 36 = -13

5. Now, we can factor the left side as a perfect square:

   (x + 6)² = -13

6. Next, we take the square root of both sides, remembering to consider both the positive and negative roots:

   x + 6 = ±√(-13)

7. We simplify this further, noting that √(-13) can be expressed in terms of imaginary numbers:

   x + 6 = ±i√13

8. Finally, isolate x by subtracting 6 from both sides:

   x = -6 ± i√13

So, the solutions to the equation x² + 12x + 49 = 0 using the method of completing the square are:

• x = -6 + i√13
• x = -6 – i√13