The value of `x^(2)+2bx+c` is positive if

Options

A. `b^(2)-4c gt 0`
B. `b^(2)-4c lt 0`
C. `c^(2) lt b`
D. `b^(2) lt c`

Answer

To determine when the value of the quadratic expression \( x^2 + 2bx + c \) is positive, we need to analyze the conditions under which this quadratic polynomial does not intersect the x-axis.

### Step-by-Step Solution:

1. **Identify the Quadratic Expression**:
The given expression is \( f(x) = x^2 + 2bx + c \).

2. **Understand the Conditions for Positivity**:
A quadratic expression \( ax^2 + bx + c \) is positive for all \( x \) if its discriminant is less than zero. This means that the quadratic does not have real roots and opens upwards.

3. **Calculate the Discriminant**:
The discriminant \( D \) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
For our expression, \( a = 1 \), \( b = 2b \), and \( c = c \). Therefore, the discriminant becomes:
\[ D = (2b)^2 - 4(1)(c) = 4b^2 - 4c \]

4. **Set the Discriminant Condition**:
For the quadratic to be positive for all \( x \), we need:
\[ D < 0 \]
This gives us:
\[ 4b^2 - 4c < 0 \]

5. **Simplify the Inequality**:
Dividing the entire inequality by 4 (since 4 is positive and does not change the direction of the inequality), we have:
\[ b^2 - c < 0 \]

6. **Final Condition**:
Rearranging gives us the condition:
\[ b^2 < c \]
This means that for the quadratic expression \( x^2 + 2bx + c \) to be positive for all \( x \), the value of \( c \) must be greater than \( b^2 \).

### Conclusion:
The value of \( x^2 + 2bx + c \) is positive if \( c > b^2 \).

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