To complete the square for the equation \( x^2 + 12x - 14 = 0 \), we can follow these steps:
-
Isolate the quadratic and linear terms: \[ x^2 + 12x = 14 \]
-
Complete the square: To complete the square, we need to add and subtract \(\left(\frac{12}{2}\right)^2\) (which is \( 36 \)) to the left side of the equation.
Therefore, we will add \( 36 \) to both sides: \[ x^2 + 12x + 36 = 14 + 36 \]
-
Rewrite the left side as a square: The left side can now be written as a perfect square: \[ (x + 6)^2 = 50 \]
-
Take the square root of both sides: \[ x + 6 = \pm \sqrt{50} \]
Since \(\sqrt{50} = 5\sqrt{2}\), we can express it more simply: \[ x + 6 = \pm 5\sqrt{2} \]
-
Solve for \(x\): Subtract \(6\) from both sides: \[ x = -6 \pm 5\sqrt{2} \]
So, the complete solution to the original equation is: \[ x = -6 + 5\sqrt{2} \quad \text{or} \quad x = -6 - 5\sqrt{2} \]
As we described the steps, the correct entries for what's missing would be:
- In the completing the square step (i.e., the blanks): "36" for both blanks.
- The square root result before solving would have necessitated \( \sqrt{50} \), which simplifies to \( 5\sqrt{2} \).
Let me know if you need any additional information!