Study the statement carefully.
Statement I : Both the roots of the equation `x^(2) - x + 1 = 0` are real.
Statement II : The roots of the equations `ax^(2) + bx + c = 0` are real if and only `b^(2) - 4ac gt 0`. Which of the following options hold?

Options

A. Both Statement I and Statement II are true
.
B. Both Statement I and Statement II are false.
C. Statement I is true but Statement II is false.
D. Statement I is false and Statement II is true.

Answer

To solve the problem, we need to analyze both statements regarding the quadratic equation \(x^2 - x + 1 = 0\) and the general form of a quadratic equation \(ax^2 + bx + c = 0\).

### Step-by-Step Solution:

**Step 1: Analyze Statement I**
- The equation given is \(x^2 - x + 1 = 0\).
- Here, \(a = 1\), \(b = -1\), and \(c = 1\).
- To determine if the roots are real, we calculate the discriminant \(D\) using the formula:
\[ D = b^2 - 4ac \]
- Substituting the values:
\[ D = (-1)^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]
- Since the discriminant \(D < 0\), this indicates that the roots are not real (they are complex).

**Conclusion for Statement I:**
- Statement I is **false** because both roots are not real.

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**Step 2: Analyze Statement II**
- The statement claims that the roots of the equation \(ax^2 + bx + c = 0\) are real if and only if \(b^2 - 4ac > 0\).
- The condition for the roots to be real is:
- If \(D > 0\), the roots are real and distinct.
- If \(D = 0\), the roots are real and equal.
- If \(D < 0\), the roots are non-real (complex).
- Therefore, the correct condition for real roots is \(D \geq 0\) (not just \(D > 0\)).

**Conclusion for Statement II:**
- Statement II is **false** because it does not account for the case when \(D = 0\).

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### Final Conclusion:
- Both Statement I and Statement II are false.