To solve the problem, let's analyze the quadratic equation given:
[tex]\[ 0 = x^2 - 4x + 5 \][/tex]
A general quadratic equation is of the form:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
Here, we can identify the coefficients as:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 5 \)[/tex]
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Let's substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into this formula:
1. Calculate [tex]\( b^2 \)[/tex]:
[tex]\[ (-4)^2 = 16 \][/tex]
2. Calculate [tex]\( 4ac \)[/tex]:
[tex]\[ 4 \cdot 1 \cdot 5 = 20 \][/tex]
3. Now, find the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = 16 - 20 = -4 \][/tex]
The value of the discriminant is [tex]\( -4 \)[/tex].
Next, we interpret the discriminant to determine the number of real solutions for the quadratic equation:
- If the discriminant [tex]\( \Delta > 0 \)[/tex], there are 2 distinct real solutions.
- If the discriminant [tex]\( \Delta = 0 \)[/tex], there is exactly 1 real solution.
- If the discriminant [tex]\( \Delta < 0 \)[/tex], there are no real solutions (but two complex solutions).
Since the discriminant here is [tex]\( -4 \)[/tex], which is less than 0, the quadratic equation has no real solutions.
Therefore, the correct answer is:
The discriminant is [tex]\(-4\)[/tex], so the equation has no real solutions.
