If a-b square = 56 and ab = 24 find the value of a2 +b2

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.

Learn how to find the value of a^2 + b^2 using the given equations (a - b)^2 = 56 and ab = 24. Step-by-step solution provided with detailed algebraic manipulation and substitution. Suitable for high school students.

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