Let's analyze the quadratic equation given by [tex]\(0 = x^2 - 4x + 5\)[/tex].
For a quadratic equation of the form [tex]\(ax^2 + bx + c\)[/tex], the discriminant is given by the expression [tex]\(b^2 - 4ac\)[/tex]. The discriminant provides information about the nature of the roots of the quadratic equation:
1. If the discriminant is greater than 0 ([tex]\(\Delta > 0\)[/tex]), the equation has two distinct real solutions.
2. If the discriminant is equal to 0 ([tex]\(\Delta = 0\)[/tex]), the equation has exactly one real solution.
3. If the discriminant is less than 0 ([tex]\(\Delta < 0\)[/tex]), the equation has no real solutions (the solutions are complex or imaginary).
In the given equation [tex]\(0 = x^2 - 4x + 5\)[/tex]:
- The coefficient [tex]\(a\)[/tex] is 1 (the coefficient of [tex]\(x^2\)[/tex]).
- The coefficient [tex]\(b\)[/tex] is -4 (the coefficient of [tex]\(x\)[/tex]).
- The constant term [tex]\(c\)[/tex] is 5.
Now let's compute the discriminant using the formula [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[ b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 5 \][/tex]
Calculating the values:
[tex]\[ (-4)^2 = 16 \][/tex]
[tex]\[ 4 \cdot 1 \cdot 5 = 20 \][/tex]
[tex]\[ b^2 - 4ac = 16 - 20 = -4 \][/tex]
The value of the discriminant is [tex]\(-4\)[/tex]. Since [tex]\(-4\)[/tex] is less than 0 ([tex]\(\Delta < 0\)[/tex]), this indicates that the quadratic equation has no real solutions. Instead, the solutions will be complex or imaginary.
Therefore, the correct interpretation is:
[tex]\[ \boxed{\text{The discriminant is } -4 \text{, so the equation has no real solutions.}} \][/tex]
