For what value of x, the following expressions are positive ?
`3x^(2)+4x+4`

Answer

To determine for what values of \( x \) the expression \( 3x^2 + 4x + 4 \) is positive, we can follow these steps:

### Step 1: Identify the coefficients
We start with the quadratic expression:
\[ f(x) = 3x^2 + 4x + 4 \]
Here, we identify the coefficients:
- \( a = 3 \)
- \( b = 4 \)
- \( c = 4 \)

### Step 2: Calculate the discriminant
The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c \) is given by:
\[ \Delta = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ \Delta = 4^2 - 4 \cdot 3 \cdot 4 \]
Calculating this gives:
\[ \Delta = 16 - 48 = -32 \]

### Step 3: Analyze the discriminant
Since the discriminant \( \Delta \) is less than 0 (\( \Delta < 0 \)), this indicates that the quadratic equation does not have real roots. Therefore, the expression \( f(x) \) does not cross the x-axis.

### Step 4: Determine the sign of the quadratic expression
Since \( a = 3 \) (the coefficient of \( x^2 \)) is positive, the parabola opens upwards. Given that there are no real roots and the parabola opens upwards, the expression \( f(x) = 3x^2 + 4x + 4 \) is positive for all real values of \( x \).

### Conclusion
Thus, the expression \( 3x^2 + 4x + 4 \) is positive for all \( x \in \mathbb{R} \).

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