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 need to analyze the behavior of the function defined by the left-hand side of the equation.

### 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 first find the derivative of \( f(x) \):
\[ f'(x) = 3x^2 - 2|a|x + 3 \]

### Step 3: Analyze the derivative
For \( f(x) \) to have only one real root, \( f'(x) \) must not change sign, meaning it should either be always positive or always negative. Since the leading coefficient (3) of \( f'(x) \) is positive, we need to ensure that \( f'(x) \) is always positive.

### Step 4: Condition for the derivative
For the quadratic \( f'(x) = 3x^2 - 2|a|x + 3 \) to be always positive, its discriminant must be less than zero:
\[ D = b^2 - 4ac < 0 \]
where \( a = 3 \), \( b = -2|a| \), and \( c = 3 \).

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

Setting the discriminant less than zero:
\[ 4|a|^2 - 36 < 0 \]

### Step 5: Solve the inequality
Rearranging gives:
\[ 4|a|^2 < 36 \]
Dividing both sides by 4:
\[ |a|^2 < 9 \]
Taking the square root:
\[ |a| < 3 \]

### Step 6: Conclusion
Thus, the values of \( a \) that satisfy this condition are:
\[ -3 < a < 3 \]
or in interval notation:
\[ a \in (-3, 3) \]

### Final Answer
The values of \( a \) such that the equation \( x^3 - |a|x^2 + 3x + 4 = 0 \) has only one real root are:
\[ a \in (-3, 3) \]