The sum of two numbers is 7 and the sum of their cubes is 133. Find the sum of their squares

Answer

To solve the problem step by step, we will use the information given about the two numbers.

### Step 1: Define the variables
Let the two numbers be \( a \) and \( b \).
We know from the problem that:
\[ a + b = 7 \]

### Step 2: Use the sum of cubes
We are also given that the sum of their cubes is:
\[ a^3 + b^3 = 133 \]

### Step 3: Use the identity for the sum of cubes
We can use the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
We can also express \( a^2 + b^2 \) in terms of \( a + b \) and \( ab \):
\[ a^2 + b^2 = (a + b)^2 - 2ab \]

### Step 4: Substitute \( a + b \) into the identity
Substituting \( a + b = 7 \) into the identity for the sum of cubes:
\[ a^3 + b^3 = (7)(a^2 - ab + b^2) \]

### Step 5: Express \( a^2 - ab + b^2 \)
We can rewrite \( a^2 - ab + b^2 \) as:
\[ a^2 - ab + b^2 = (a^2 + b^2) - ab \]

### Step 6: Substitute into the sum of cubes equation
Thus, we can write:
\[ 133 = 7((a^2 + b^2) - ab) \]

### Step 7: Solve for \( ab \)
Now we need to find \( ab \). First, we will rearrange the equation:
\[ 133 = 7(a^2 + b^2 - ab) \]
Dividing both sides by 7:
\[ \frac{133}{7} = a^2 + b^2 - ab \]
Calculating \( \frac{133}{7} \):
\[ 19 = a^2 + b^2 - ab \]

### Step 8: Use \( a + b \) to find \( ab \)
We also know:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
Substituting \( a + b = 7 \):
\[ 7^2 = a^2 + 2ab + b^2 \]
This gives:
\[ 49 = a^2 + 2ab + b^2 \]

### Step 9: Substitute \( a^2 + b^2 \) into the equation
Now we can express \( a^2 + b^2 \) in terms of \( ab \):
\[ a^2 + b^2 = 49 - 2ab \]

### Step 10: Substitute back into the previous equation
Now we substitute \( a^2 + b^2 \) into the equation we derived earlier:
\[ 19 = (49 - 2ab) - ab \]
This simplifies to:
\[ 19 = 49 - 3ab \]
Rearranging gives:
\[ 3ab = 49 - 19 \]
\[ 3ab = 30 \]
Thus:
\[ ab = 10 \]

### Step 11: Find \( a^2 + b^2 \)
Now we can find \( a^2 + b^2 \):
\[ a^2 + b^2 = 49 - 2ab \]
Substituting \( ab = 10 \):
\[ a^2 + b^2 = 49 - 2(10) \]
\[ a^2 + b^2 = 49 - 20 = 29 \]

### Conclusion
The sum of the squares of the two numbers is:
\[ \boxed{29} \]