If the sum of two numbers is 5 and the sum of their cubes is 35, find the sum of their squares

Answer

To solve the problem step by step, we will use the information given in the question and apply relevant algebraic formulas.

### Step 1: Define the variables
Let the two numbers be \( a \) and \( b \).

### Step 2: Write down the equations
From the problem, we know:
1. The sum of the two numbers:
\[ a + b = 5 \quad \text{(Equation 1)} \]
2. The sum of their cubes:
\[ a^3 + b^3 = 35 \quad \text{(Equation 2)} \]

### Step 3: Use the identity for the sum of cubes
We can use the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Substituting \( a + b = 5 \) into the equation:
\[ a^3 + b^3 = 5(a^2 - ab + b^2) \]
Setting this equal to 35 (from Equation 2):
\[ 5(a^2 - ab + b^2) = 35 \]
Dividing both sides by 5:
\[ a^2 - ab + b^2 = 7 \quad \text{(Equation 3)} \]

### Step 4: Use the identity for the sum of squares
We also know that:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting \( a + b = 5 \):
\[ a^2 + b^2 = 5^2 - 2ab \]
This simplifies to:
\[ a^2 + b^2 = 25 - 2ab \quad \text{(Equation 4)} \]

### Step 5: Relate Equations 3 and 4
From Equation 3, we can express \( a^2 + b^2 \):
\[ a^2 + b^2 = 7 + ab \quad \text{(from rearranging Equation 3)} \]

### Step 6: Set the two expressions for \( a^2 + b^2 \) equal
Now we have two expressions for \( a^2 + b^2 \):
1. From Equation 4: \( a^2 + b^2 = 25 - 2ab \)
2. From Equation 3: \( a^2 + b^2 = 7 + ab \)

Setting them equal:
\[ 25 - 2ab = 7 + ab \]

### Step 7: Solve for \( ab \)
Rearranging gives:
\[ 25 - 7 = 2ab + ab \]
\[ 18 = 3ab \]
Dividing both sides by 3:
\[ ab = 6 \quad \text{(Equation 5)} \]

### Step 8: Find \( a^2 + b^2 \)
Now substitute \( ab = 6 \) back into either expression for \( a^2 + b^2 \). Using Equation 4:
\[ a^2 + b^2 = 25 - 2(6) \]
\[ a^2 + b^2 = 25 - 12 \]
\[ a^2 + b^2 = 13 \]

### Final Answer
Thus, the sum of the squares of the two numbers is:
\[ \boxed{13} \]