The quadratic equation `2x^(2) + x + 4 ` has ………real roots

Answer

To determine whether the quadratic equation \(2x^2 + x + 4\) has real roots, we will use the discriminant method. The discriminant \(D\) of a quadratic equation in the form \(ax^2 + bx + c\) is given by the formula:

\[ D = b^2 - 4ac \]

### Step-by-Step Solution:

1. **Identify coefficients**:
- From the equation \(2x^2 + x + 4\), we identify:
- \(a = 2\)
- \(b = 1\)
- \(c = 4\)

2. **Calculate the discriminant**:
- Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula:
\[ D = b^2 - 4ac \]
- Plugging in the values:
\[ D = (1)^2 - 4 \cdot (2) \cdot (4) \]
- Calculate \(b^2\):
\[ D = 1 - 4 \cdot 2 \cdot 4 \]
- Calculate \(4ac\):
\[ D = 1 - 32 \]
- Finally, simplify:
\[ D = 1 - 32 = -31 \]

3. **Determine the nature of the roots**:
- The discriminant \(D = -31\) is less than zero.
- According to the rules of quadratic equations:
- If \(D > 0\), there are two distinct real roots.
- If \(D = 0\), there is exactly one real root (a repeated root).
- If \(D < 0\), there are no real roots.
- Since \(D = -31 < 0\), we conclude that the quadratic equation \(2x^2 + x + 4\) has no real roots.

### Final Answer:
The quadratic equation \(2x^2 + x + 4\) has **no real roots**.

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