*x*(*hx* + 72) = 24

In the given equation, *h* is a constant. If the equation has exactly one real solution, what is the value of *h*?

The given equation is:

\[
x(hx + 72) = 24
\]

Expanding this equation:

\[
hx^2 + 72x = 24
\]

This can be rewritten as a standard quadratic equation:

\[
hx^2 + 72x - 24 = 0
\]

For a quadratic equation \(ax^2 + bx + c = 0\) to have exactly one real solution, its discriminant must be zero. The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:

\[
\Delta = b^2 - 4ac
\]

In our equation, \(a = h\), \(b = 72\), and \(c = -24\). Plugging these into the discriminant formula:

\[
\Delta = 72^2 - 4(h)(-24)
\]

Simplifying:

\[
\Delta = 5184 + 96h
\]

For there to be exactly one real solution, the discriminant must be zero:

\[
5184 + 96h = 0
\]

Solving for \(h\):

\[
96h = -5184
\]
\[
h = \frac{-5184}{96}
\]
\[
h = -54
\]

### Answer:
The value of \(h\) is \( \boxed{-54} \).

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Do you need any more details about the steps?

Here are some related questions you might consider:
1. What if the discriminant was positive? What would it imply?
2. Can you find the roots of the equation when \( h = -54 \)?
3. How does changing the coefficient \( h \) affect the nature of the roots?
4. What would the equation look like if there were no real solutions?
5. How does the quadratic formula relate to the discriminant?

**Tip:** Always remember, a zero discriminant indicates that the quadratic has exactly one real solution, implying the vertex of the parabola touches the x-axis.

Learn how to solve the quadratic equation x(hx + 72) = 24 for a constant h. This problem covers key algebraic concepts, including quadratic equations, discriminant, and quadratic formula theorem. Suitable for high school students (Grades 10-12).

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