Algebraic Expression Calculation

This problem requires us to use fundamental algebraic identities to find the value of a given expression. We are provided with the values of \((a^2 + b^2)\) and \((a \cdot b)\), and our goal is to determine the value of \((a + b) \div (a - b)\).

Understanding the Key Algebraic Identities

To solve this problem, we will use two common algebraic identities:

• The square of a sum: \((a + b)^2 = a^2 + b^2 + 2ab\)
• The square of a difference: \((a - b)^2 = a^2 + b^2 - 2ab\)

Step-by-Step Solution

Step 1: Finding the Value of \((a + b)\)

We are given that \(a^2 + b^2 = 41\) and \(a \cdot b = 20\). Let's use the identity for \((a + b)^2\):

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

Substitute the given values into the identity:

$$(a + b)^2 = 41 + 2(20)$$ $$(a + b)^2 = 41 + 40$$ $$(a + b)^2 = 81$$

Now, take the square root of both sides to find \((a + b)\):

$$(a + b) = \pm\sqrt{81}$$ $$(a + b) = \pm 9$$

Step 2: Finding the Value of \((a - b)\)

Next, let's use the identity for \((a - b)^2\):

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

Substitute the given values into this identity:

$$(a - b)^2 = 41 - 2(20)$$ $$(a - b)^2 = 41 - 40$$ $$(a - b)^2 = 1$$

Now, take the square root of both sides to find \((a - b)\):

$$(a - b) = \pm\sqrt{1}$$ $$(a - b) = \pm 1$$

Step 3: Calculating \((a + b) \div (a - b)\)

We need to find the ratio \((a + b) \div (a - b)\). We have found that \((a + b) = \pm 9\) and \((a - b) = \pm 1\). Let's consider the combinations that yield a positive result, as the given options are positive integers.

• Case 1: If \((a + b) = 9\) and \((a - b) = 1\), then: $$(a + b) \div (a - b) = 9 \div 1 = 9$$
• Case 2: If \((a + b) = -9\) and \((a - b) = -1\), then: $$(a + b) \div (a - b) = (-9) \div (-1) = 9$$

In both cases that result in a positive value, the answer is 9. We can verify this with specific values for \(a\) and \(b\). If \(a+b=9\) and \(a-b=1\):
Adding the two equations: \(2a = 10 \implies a = 5\)
Subtracting the two equations: \(2b = 8 \implies b = 4\)
Check: \(a^2 + b^2 = 5^2 + 4^2 = 25 + 16 = 41\) (Correct)
Check: \(a \cdot b = 5 \cdot 4 = 20\) (Correct)
So, when \(a=5\) and \(b=4\), \((a+b)/(a-b) = (5+4)/(5-4) = 9/1 = 9\).

Therefore, the value of \((a + b) \div (a - b)\) is 9.

Final Answer Summary

The result aligns with the options provided.