If `a^(3) + b^(3) = 218 and a + b = 2`, then the value of ab is :

Options

A. 34
B. -31
C. -35
D. 32

Answer

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

1. \( a^3 + b^3 = 218 \)
2. \( a + b = 2 \)

We want to find the value of \( ab \).

### Step 1: Use the identity for the sum of cubes
We know that the sum of cubes can be expressed as:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
From the second equation, we have \( a + b = 2 \).

### Step 2: Substitute \( a + b \) into the identity
Substituting \( a + b \) into the identity gives us:
\[ a^3 + b^3 = 2(a^2 - ab + b^2) \]

### Step 3: Express \( a^2 + b^2 \) in terms of \( ab \)
We can express \( a^2 + b^2 \) using the identity:
\[ a^2 + b^2 = (a + b)^2 - 2ab \]
Substituting \( a + b = 2 \) into this gives:
\[ a^2 + b^2 = 2^2 - 2ab = 4 - 2ab \]

### Step 4: Substitute \( a^2 + b^2 \) back into the equation
Now we substitute \( a^2 + b^2 \) back into the equation for \( a^3 + b^3 \):
\[ a^3 + b^3 = 2((4 - 2ab) - ab) = 2(4 - 3ab) \]
So we have:
\[ a^3 + b^3 = 8 - 6ab \]

### Step 5: Set the equation equal to 218
Now we set this equal to the first equation:
\[ 8 - 6ab = 218 \]

### Step 6: Solve for \( ab \)
Rearranging gives:
\[ -6ab = 218 - 8 \]
\[ -6ab = 210 \]
\[ ab = -\frac{210}{6} = -35 \]

Thus, the value of \( ab \) is:
\[ \boxed{-35} \]