The equation you've given is:

$A^2 = (b - a)(p - a)$

This equation expresses a relationship between $A$, $a$, $b$, and $p$. To understand or manipulate it further, we can approach it in different ways, depending on what you're looking for. For example, we could:

1. Solve for $A$: Take the square root of both sides to isolate $A$.

$A = \sqrt{(b - a)(p - a)}$

2. Expand the right-hand side: If you want to expand it as a quadratic expression in terms of $a$, you can distribute the terms:

$(b - a)(p - a) = bp - ab - ap + a^2$

So, the equation becomes:

$A^2 = bp - ab - ap + a^2$

3. Analyze for specific values: If you have numerical values for $b$, $a$, and $p$, you could substitute them in to find a value for $A$.

Could you specify what you would like to do with this equation? For instance, are you trying to isolate one of the variables, solve for specific values, or understand its geometric meaning?

Further Exploration:

1. How would you isolate $a$ from this equation?
2. What happens if we set $a = b$ or $a = p$?
3. How can this equation be applied in a geometric context, such as area or distance problems?
4. Can this equation be interpreted as a product of two linear factors? If so, how?
5. What is the meaning of $A$ in this equation—could it represent a distance, an area, or something else?

Tip:

When working with quadratic expressions or equations involving squares, always consider whether completing the square or factoring could simplify the problem.