The equation given is:

\[
h^2 + 74 = 0
\]

To solve for \( h \), let's isolate \( h^2 \) by subtracting 74 from both sides:

\[
h^2 = -74
\]

Now, to solve for \( h \), take the square root of both sides:

\[
h = \pm \sqrt{-74}
\]

Since the square root of a negative number involves imaginary numbers, we can express this as:

\[
h = \pm \sqrt{74}i
\]

where \( i \) is the imaginary unit (\( i = \sqrt{-1} \)).

Therefore, the solutions for \( h \) are:

\[
h = \sqrt{74}i \quad \text{and} \quad h = -\sqrt{74}i
\]

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. What is the meaning of the imaginary unit \( i \)?
2. How do you simplify square roots of negative numbers?
3. What are complex numbers, and how do they relate to imaginary numbers?
4. How can you represent the solutions on the complex plane?
5. What are the properties of quadratic equations with complex solutions?

**Tip:** Remember, the square root of a negative number always results in an imaginary number.

Learn how to solve the quadratic equation h^2 + 74 = 0 using imaginary numbers. This solution explores key algebraic concepts and introduces complex numbers for students in Grades 9-12.

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