If `x ^(3) + 3x ^(2) + 3x = 7,` then the x is equal to

Options

A. 2
B. `sqrt6`
C. 1
D. `-1`

Answer

To solve the equation \( x^3 + 3x^2 + 3x = 7 \), we will follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ x^3 + 3x^2 + 3x = 7 \]
We can rearrange this by moving 7 to the left side:
\[ x^3 + 3x^2 + 3x - 7 = 0 \]

### Step 2: Add 1 to both sides
Next, we will add 1 to both sides of the equation:
\[ x^3 + 3x^2 + 3x + 1 = 7 + 1 \]
This simplifies to:
\[ x^3 + 3x^2 + 3x + 1 = 8 \]

### Step 3: Recognize the left side as a cube
The left side can be factored as the cube of a binomial:
\[ (x + 1)^3 = 8 \]

### Step 4: Take the cube root of both sides
Now, we take the cube root of both sides:
\[ x + 1 = \sqrt[3]{8} \]
Since \( \sqrt[3]{8} = 2 \), we have:
\[ x + 1 = 2 \]

### Step 5: Solve for x
Now, we will solve for \( x \) by subtracting 1 from both sides:
\[ x = 2 - 1 \]
Thus, we find:
\[ x = 1 \]

### Final Answer
The value of \( x \) is \( 1 \).

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