Fill in the steps to complete the square: x2 +12x − 14 = 0 x2 + 12x = 14 x2 +12x + Response area = 14+ Response area We factor and simplify to get: (x+6)2 = Response area At this point, we need to Response area After that is done, we will have: Response area = Response area We get isolate the variable by: Response area Final answers are: Response area

Answer 1

Let's complete the square step by step for the equation \( x^2 + 12x - 14 = 0 \).

1. Starting with the original equation: \[ x^2 + 12x - 14 = 0 \]

2. Move the constant term to the other side: \[ x^2 + 12x = 14 \]

3. Identify the coefficient of \( x \), divide by 2, and square it:

   • Coefficient of \( x \) is 12.
   • Half of 12 is 6, and \( 6^2 = 36 \).

4. Add the squared term to both sides: \[ x^2 + 12x + 36 = 14 + 36 \]

5. Simplify the right side: \[ x^2 + 12x + 36 = 50 \]

6. Factor the left side: \[ (x + 6)^2 = 50 \]

7. Take the square root of both sides:

   • Remember to consider both the positive and negative roots. \[ x + 6 = \pm \sqrt{50} \]

8. Isolate the variable \( x \): \[ x = -6 \pm \sqrt{50} \]

9. Final answers:

   • Since \( \sqrt{50} = 5\sqrt{2} \), we can rewrite the answers as: \[ x = -6 + 5\sqrt{2} \quad \text{or} \quad x = -6 - 5\sqrt{2} \]

So, filled in response areas would be:

• Response area 1: \( 36 \)
• Response area 2: \( 36 \)
• Response area 3: \( \pm \sqrt{50} \)
• Response area 4: \( -6 \pm \sqrt{50} \)
• Response area 5: \( -6 \pm 5\sqrt{2} \)