We are given the equation:

\[
k^2 - 14k - 7 = -76
\]

To solve for \(k\), follow these steps:

### Step 1: Move all terms to one side of the equation
We start by adding 76 to both sides to get a standard quadratic equation:

\[
k^2 - 14k - 7 + 76 = 0
\]

Simplify:

\[
k^2 - 14k + 69 = 0
\]

### Step 2: Use the quadratic formula
The quadratic formula is:

\[
k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \(k^2 - 14k + 69 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -14\)
- \(c = 69\)

### Step 3: Calculate the discriminant
The discriminant is:

\[
\Delta = b^2 - 4ac = (-14)^2 - 4(1)(69) = 196 - 276 = -80
\]

Since the discriminant is negative, the solutions will be complex numbers.

### Step 4: Solve for \(k\)
Substitute the values into the quadratic formula:

\[
k = \frac{-(-14) \pm \sqrt{-80}}{2(1)} = \frac{14 \pm \sqrt{-80}}{2}
\]

Simplify:

\[
k = \frac{14 \pm \sqrt{80}i}{2} = \frac{14 \pm 4\sqrt{5}i}{2}
\]

Now, simplify further:

\[
k = 7 \pm 2\sqrt{5}i
\]

### Final Answer:
The solutions are:

\[
k = 7 + 2\sqrt{5}i \quad \text{or} \quad k = 7 - 2\sqrt{5}i
\]

These are the two complex solutions.

Would you like further details on any step, or have any other questions?

Here are five related questions:
1. What are the conditions for a quadratic equation to have real solutions?
2. How do you interpret complex solutions in real-world applications?
3. Can you explain how to simplify square roots of negative numbers?
4. What is the significance of the discriminant in quadratic equations?
5. How would the solution change if the discriminant were positive?

**Tip**: When solving quadratic equations, always check the discriminant to determine the nature of the roots: real or complex.

Learn how to solve the quadratic equation k^2 - 14k + 69 = 0, which involves complex solutions due to a negative discriminant. This detailed solution covers the quadratic formula and simplifying complex numbers. Suitable for students in Grades 9-12.

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