To determine the value of the discriminant for the given quadratic equation [tex]\( x^2 - 4x + 5 = 0 \)[/tex] and analyze the number of real solutions, let's proceed step by step.
### Step 1: Identify the coefficients
The standard form of a quadratic equation is:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
For the given equation [tex]\( x^2 - 4x + 5 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 5 \)[/tex]
### Step 2: Compute the discriminant
The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Now substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = (-4)^2 - 4 \cdot 1 \cdot 5 \][/tex]
[tex]\[ \Delta = 16 - 20 \][/tex]
[tex]\[ \Delta = -4 \][/tex]
### Step 3: Interpret the discriminant
The discriminant tells us the nature of the roots (real or complex) of the quadratic equation:
- If [tex]\( \Delta > 0 \)[/tex], the equation has two distinct real solutions.
- If [tex]\( \Delta = 0 \)[/tex], the equation has exactly one real solution (a repeated root).
- If [tex]\( \Delta < 0 \)[/tex], the equation has no real solutions, but two complex solutions.
Since [tex]\( \Delta = -4 \)[/tex] and [tex]\( -4 < 0 \)[/tex], this indicates that the quadratic equation [tex]\( x^2 - 4x + 5 = 0 \)[/tex] has no real solutions. Instead, it will have two complex solutions.
### Conclusion
Given the computed discriminant and its interpretation:
Option 2 is correct:
"The discriminant is -4, so the equation has no real solutions."
