If `x + y = 14, x^(3) + y^(3) = 1064`, then the value of `(x-y)^(2)` is:

Options

A. 36
B. 64
C. 81
D. 100

Answer

To solve the problem step by step, we start with the given equations:

1. **Given Equations:**
\[ x + y = 14 \]
\[ x^3 + y^3 = 1064 \]

2. **Using the Formula for the Sum of Cubes:**
The formula for the sum of cubes is:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
Substituting the known value of \(x + y\):
\[ 1064 = 14(x^2 - xy + y^2) \]

3. **Solving for \(x^2 - xy + y^2\):**
Dividing both sides by 14:
\[ x^2 - xy + y^2 = \frac{1064}{14} = 76 \]

4. **Using the Identity for \(x^2 + y^2\):**
We know that:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Substituting \(x + y = 14\):
\[ x^2 + y^2 = 14^2 - 2xy = 196 - 2xy \]

5. **Substituting into the Equation:**
Now, we can express \(x^2 - xy + y^2\) in terms of \(xy\):
\[ x^2 - xy + y^2 = (x^2 + y^2) - xy = (196 - 2xy) - xy = 196 - 3xy \]
Setting this equal to 76:
\[ 196 - 3xy = 76 \]

6. **Solving for \(xy\):**
Rearranging gives:
\[ 3xy = 196 - 76 = 120 \]
Therefore:
\[ xy = \frac{120}{3} = 40 \]

7. **Finding \(x^2 + y^2\):**
Now substitute \(xy\) back into the equation for \(x^2 + y^2\):
\[ x^2 + y^2 = 196 - 2(40) = 196 - 80 = 116 \]

8. **Finding \(x - y\):**
We can use the identity:
\[ (x - y)^2 = (x + y)^2 - 4xy \]
Substituting the known values:
\[ (x - y)^2 = 14^2 - 4(40) = 196 - 160 = 36 \]

9. **Final Result:**
Thus, the value of \((x - y)^2\) is:
\[ \boxed{36} \]