Solving for a^2 + b^2 using Algebraic Identities

This problem asks us to find the value of \(a^2 + b^2\) given two pieces of information about variables \(a\) and \(b\): their difference \(a - b\) and their product \(ab\). We are given that \(a - b = 5\) and \(ab = 24\). To solve this, we can use a common algebraic identity.

Understanding the Relationship: (a - b)^2

A key algebraic identity relates the difference of two terms, their squares, and their product. This identity is: \[ (a - b)^2 = a^2 - 2ab + b^2 \] Notice that the term \(a^2 + b^2\) appears in this identity. We can rearrange this identity to solve for \(a^2 + b^2\).

Rearranging the Identity to Find a^2 + b^2

Let's rearrange the identity \( (a - b)^2 = a^2 - 2ab + b^2 \) to isolate the term we want to find, \(a^2 + b^2\). We can do this by adding \(2ab\) to both sides of the equation: \[ (a - b)^2 + 2ab = a^2 - 2ab + b^2 + 2ab \] \[ (a - b)^2 + 2ab = a^2 + b^2 \] So, we have the formula: \[ a^2 + b^2 = (a - b)^2 + 2ab \]

Substituting Given Values and Calculating

Now we can use the values provided in the question. We know that \(a - b = 5\) and \(ab = 24\). We substitute these values into the rearranged formula for \(a^2 + b^2\):

Substitute \(a - b = 5\): \[ a^2 + b^2 = (5)^2 + 2ab \] Substitute \(ab = 24\): \[ a^2 + b^2 = (5)^2 + 2(24) \]

Now, perform the calculations: \[ a^2 + b^2 = 25 + 48 \] \[ a^2 + b^2 = 73 \]

Thus, the value of \(a^2 + b^2\) is 73.

Step-by-Step Solution Summary

Here are the steps we followed:

• Identify the given information: \(a - b = 5\) and \(ab = 24\).
• Recall or derive the algebraic identity relating \(a^2 + b^2\), \(a - b\), and \(ab\).
• The identity is \( (a - b)^2 = a^2 - 2ab + b^2 \).
• Rearrange the identity to get \( a^2 + b^2 = (a - b)^2 + 2ab \).
• Substitute the given values: \( a^2 + b^2 = (5)^2 + 2(24) \).
• Calculate the result: \( a^2 + b^2 = 25 + 48 = 73 \).

Comparing with Options

The calculated value for \(a^2 + b^2\) is 73. Let's check the given options:

• Option 1: 73
• Option 2: 36
• Option 3: 1
• Option 4: 72

Our calculated value matches Option 1.

Additional Information: Other Related Identities

Besides \( (a - b)^2 \), there are other useful algebraic identities involving squares and products of terms. Understanding these can help solve similar problems.

• Sum of Squares: \( a^2 + b^2 \) can also be related to \( (a+b)^2 \) and \( ab \):

  \[ (a + b)^2 = a^2 + 2ab + b^2 \] \[ a^2 + b^2 = (a + b)^2 - 2ab \] If we were given \(a+b\) and \(ab\), we would use this version.

• Difference of Squares:

  \[ a^2 - b^2 = (a - b)(a + b) \] This identity is useful when factoring or dealing with differences of squared terms.

These identities are fundamental tools in algebra for simplifying expressions and solving equations.