We are given the following information:

$(a - b)^2 = 56 \quad \text{and} \quad ab = 24.$

We are asked to find the value of $a^2 + b^2$.

Step 1: Expand $(a - b)^2$

We know the formula for expanding $(a - b)^2$:

$(a - b)^2 = a^2 - 2ab + b^2.$

Given that $(a - b)^2 = 56$ and $ab = 24$, substitute these values into the equation:

$a^2 - 2ab + b^2 = 56.$

Substitute $ab = 24$ into the equation:

$a^2 - 2(24) + b^2 = 56.$

Simplify:

$a^2 - 48 + b^2 = 56.$

Step 2: Solve for $a^2 + b^2$

Add 48 to both sides of the equation:

$a^2 + b^2 = 56 + 48 = 104.$

Thus, the value of $a^2 + b^2$ is:

$\boxed{104}.$

Do you want further details or have any questions?

Here are 5 related questions you could explore:

1. How do you find $a + b$ given $ab$ and $a^2 + b^2$?
2. What is the geometric significance of $a^2 + b^2$ in terms of vectors?
3. Can you find $a + b$ if $a^2 + b^2 = 104$ and $ab = 24$?
4. What are the possible values of $a$ and $b$ given the conditions?
5. How does the discriminant of a quadratic equation relate to these types of problems?

Tip: When dealing with expressions like $(a - b)^2$, always remember to expand them using algebraic identities.