Let `'m'` be a real number, and suppose that two of the three solutions of the cubic equation `x^(3)+3x^(2)-34x=m` differ by `1`. Then possible value of `'m'` is/are

Options

A. (a) `120`
B. (b) `80`
C. (c) `-48`
D. (d) `-32`

Explanation

To solve the problem, we start with the cubic equation:

\[ x^3 + 3x^2 - 34x = m \]

We know that two of the roots of this equation differ by 1. Let's denote these roots as \( a \) and \( a + 1 \). The third root will be denoted as \( b \).

### Step 1: Set up the equations for the roots

Since \( a \) and \( a + 1 \) are roots, we can substitute them into the cubic equation:

1. For \( x = a \):
\[ a^3 + 3a^2 - 34a = m \tag{1} \]

2. For \( x = a + 1 \):
\[ (a + 1)^3 + 3(a + 1)^2 - 34(a + 1) = m \tag{2} \]

### Step 2: Expand equation (2)

Now, we expand equation (2):

\[ (a + 1)^3 = a^3 + 3a^2 + 3a + 1 \]
\[ 3(a + 1)^2 = 3(a^2 + 2a + 1) = 3a^2 + 6a + 3 \]
\[ -34(a + 1) = -34a - 34 \]

Combining these, we have:

\[ m = a^3 + 3a^2 + 3a + 1 + 3a^2 + 6a + 3 - 34a - 34 \]
\[ = a^3 + 6a^2 - 25a - 30 \tag{3} \]

### Step 3: Set equations (1) and (3) equal to each other

Since both equations equal \( m \), we can set them equal to each other:

\[ a^3 + 3a^2 - 34a = a^3 + 6a^2 - 25a - 30 \]

### Step 4: Simplify the equation

Subtract \( a^3 \) from both sides:

\[ 3a^2 - 34a = 6a^2 - 25a - 30 \]

Rearranging gives:

\[ 0 = 6a^2 - 25a - 30 - 3a^2 + 34a \]
\[ 0 = 3a^2 + 9a - 30 \]

### Step 5: Factor the quadratic equation

We can factor out a 3:

\[ 0 = a^2 + 3a - 10 \]

This can be factored as:

\[ 0 = (a - 2)(a + 5) \]

### Step 6: Solve for \( a \)

Setting each factor to zero gives us the possible values for \( a \):

1. \( a - 2 = 0 \) → \( a = 2 \)
2. \( a + 5 = 0 \) → \( a = -5 \)

### Step 7: Find the corresponding values of \( m \)

Now we substitute these values of \( a \) back into equation (1) to find \( m \).

1. For \( a = 2 \):
\[ m = 2^3 + 3(2^2) - 34(2) \]
\[ = 8 + 12 - 68 = -48 \]

2. For \( a = -5 \):
\[ m = (-5)^3 + 3(-5)^2 - 34(-5) \]
\[ = -125 + 75 + 170 = 120 \]

### Conclusion

The possible values of \( m \) are:

\[ m = -48 \quad \text{and} \quad m = 120 \]