To determine the nature of the roots of the quadratic equation
[tex]\[ 2x^2 - x + 1 = 0 \][/tex],
we follow these steps:
1. Identify the coefficients:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -1\)[/tex]
- [tex]\(c = 1\)[/tex]
2. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
Substituting the coefficients into the discriminant formula:
[tex]\[
\Delta = (-1)^2 - 4 \cdot 2 \cdot 1
\][/tex]
3. Compute the discriminant:
[tex]\[
\Delta = 1 - 8 = -7
\][/tex]
4. Determine the nature of the roots based on the discriminant:
- If [tex]\(\Delta > 0\)[/tex], the equation has two distinct real roots:
- If [tex]\(\Delta\)[/tex] is a perfect square, the roots are rational (Option B).
- If [tex]\(\Delta\)[/tex] is not a perfect square, the roots are irrational (Option C).
- If [tex]\(\Delta = 0\)[/tex], the equation has exactly one real double root (Option A).
- If [tex]\(\Delta < 0\)[/tex], the equation has two complex (non-real) roots (Option D).
Since the discriminant [tex]\(\Delta = -7\)[/tex] is less than zero ([tex]\(\Delta < 0\)[/tex]),
The nature of the roots of the quadratic equation [tex]\(2x^2 - x + 1 = 0\)[/tex] is:
D. Two non-real roots
