Find the minimum value of the quadratic expression `4x^(2) -3x + 4`.

Options

A. `(-55)/(16)`
B. `(55)/(16)`
C. `(16)/(15)`
D. `(161)/(22)`

Explanation

To find the minimum value of the quadratic expression \(4x^2 - 3x + 4\), we can follow these steps:

### Step 1: Identify the coefficients
The quadratic expression is in the standard form \(ax^2 + bx + c\). Here, we have:
- \(a = 4\)
- \(b = -3\)
- \(c = 4\)

### Step 2: Calculate the discriminant
The discriminant \(D\) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \(a\), \(b\), and \(c\):
\[ D = (-3)^2 - 4 \cdot 4 \cdot 4 \]
Calculating this:
\[ D = 9 - 64 = -55 \]

### Step 3: Determine if the quadratic has a minimum value
Since \(a > 0\) (specifically \(a = 4\)), the parabola opens upwards, and thus there is a minimum value.

### Step 4: Calculate the minimum value using the vertex formula
The x-coordinate of the vertex (which gives the minimum value) can be found using:
\[ x = -\frac{b}{2a} \]
Substituting the values of \(b\) and \(a\):
\[ x = -\frac{-3}{2 \cdot 4} = \frac{3}{8} \]

### Step 5: Substitute \(x\) back into the quadratic expression
Now, we substitute \(x = \frac{3}{8}\) back into the original expression to find the minimum value:
\[ f\left(\frac{3}{8}\right) = 4\left(\frac{3}{8}\right)^2 - 3\left(\frac{3}{8}\right) + 4 \]
Calculating each term:
\[ = 4 \cdot \frac{9}{64} - \frac{9}{8} + 4 \]
\[ = \frac{36}{64} - \frac{72}{64} + \frac{256}{64} \]
Combining these:
\[ = \frac{36 - 72 + 256}{64} = \frac{220}{64} = \frac{55}{16} \]

### Final Answer
The minimum value of the quadratic expression \(4x^2 - 3x + 4\) is:
\[ \frac{55}{16} \]