Let's answer the question step-by-step:
1. The quadratic equation given is [tex]\(0 = x^2 - 4x + 5\)[/tex].
2. The general form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. In this case, [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = 5 \)[/tex].
3. The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the formula [tex]\( \Delta = b^2 - 4ac \)[/tex].
4. Substitute the values [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = 5 \)[/tex] into the discriminant formula:
[tex]\[ \Delta = (-4)^2 - 4(1)(5) \][/tex]
[tex]\[ \Delta = 16 - 20 \][/tex]
[tex]\[ \Delta = -4 \][/tex]
5. The value of the discriminant, [tex]\( \Delta \)[/tex], is -4.
6. The discriminant indicates the nature of the roots of the quadratic equation:
- If [tex]\( \Delta > 0 \)[/tex], there are two distinct real solutions.
- If [tex]\( \Delta = 0 \)[/tex], there is exactly one real solution.
- If [tex]\( \Delta < 0 \)[/tex], there are no real solutions.
7. Since the discriminant is -4, which is less than 0, the quadratic equation has no real solutions.
### Conclusion
The discriminant is -4, so the equation has no real solutions.
Therefore, the correct answer is:
- The discriminant is -4, so the equation has no real solutions.
