Find the value of such that `x^(3)-|a|x^(2)+ 3x +4 = 0` has only one real root.

Answer

To find the value of \( a \) such that the equation \( x^3 - |a|x^2 + 3x + 4 = 0 \) has only one real root, we will follow these steps:

### Step 1: Define the function
Let \( f(x) = x^3 - |a|x^2 + 3x + 4 \).

### Step 2: Find the derivative
To determine the nature of the roots, we need to find the derivative of \( f(x) \):
\[ f'(x) = 3x^2 - 2|a|x + 3 \]

### Step 3: Analyze the derivative
For the function \( f(x) \) to have only one real root, the derivative \( f'(x) \) must not change sign (it should be either always positive or always negative). This means that the quadratic equation \( 3x^2 - 2|a|x + 3 \) should have no real roots.

### Step 4: Apply the condition for no real roots
A quadratic equation \( Ax^2 + Bx + C \) has no real roots if the discriminant \( D = B^2 - 4AC < 0 \). Here, we have:
- \( A = 3 \)
- \( B = -2|a| \)
- \( C = 3 \)

So, the discriminant is:
\[ D = (-2|a|)^2 - 4 \cdot 3 \cdot 3 \]
\[ D = 4|a|^2 - 36 \]

### Step 5: Set the discriminant less than zero
To ensure that there are no real roots, we set the discriminant less than zero:
\[ 4|a|^2 - 36 < 0 \]

### Step 6: Solve the inequality
Rearranging the inequality gives:
\[ 4|a|^2 < 36 \]
\[ |a|^2 < 9 \]
\[ |a| < 3 \]

### Step 7: Conclusion
The values of \( a \) that satisfy this condition are:
\[ -3 < a < 3 \]

Thus, the final answer is:
\[ a \in (-3, 3) \]