The minimum value of `2x^2 +x-1` is

Options

A. `-(1)/(4)`
B. `(3)/(4)`
C. `-(9)/(8)`
D. `(9)/(4)`

Explanation

To find the minimum value of the quadratic expression \(2x^2 + x - 1\), we can use the formula for the minimum value of a quadratic function \(y = ax^2 + bx + c\), where \(a > 0\). The minimum value is given by:

\[ \text{Minimum value of } y = -\frac{b^2 - 4ac}{4a} \]

### Step 1: Identify the coefficients
In the expression \(2x^2 + x - 1\), we identify the coefficients:
- \(a = 2\)
- \(b = 1\)
- \(c = -1\)

### Step 2: Substitute the coefficients into the formula
Now, we substitute \(a\), \(b\), and \(c\) into the formula:

\[ \text{Minimum value of } y = -\frac{(1)^2 - 4(2)(-1)}{4(2)} \]

### Step 3: Calculate \(b^2 - 4ac\)
Next, we calculate \(b^2 - 4ac\):

\[ b^2 = (1)^2 = 1 \]
\[ 4ac = 4 \times 2 \times (-1) = -8 \]
\[ b^2 - 4ac = 1 - (-8) = 1 + 8 = 9 \]

### Step 4: Substitute back into the formula
Now, substitute \(b^2 - 4ac\) back into the formula:

\[ \text{Minimum value of } y = -\frac{9}{4(2)} = -\frac{9}{8} \]

### Step 5: Conclusion
Thus, the minimum value of the expression \(2x^2 + x - 1\) is:

\[ \boxed{-\frac{9}{8}} \]