To determine the discriminant and the number of real solutions for the quadratic equation [tex]\( 0 = x^2 - 4x + 5 \)[/tex], we follow these steps:
1. Identify the coefficients: The quadratic equation is in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here, the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 5 \)[/tex]
2. Calculate 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]
Plugging the coefficients into the formula, we get:
[tex]\[
\Delta = (-4)^2 - 4 \cdot 1 \cdot 5
\][/tex]
3. Simplify the discriminant:
[tex]\[
\Delta = 16 - 20 = -4
\][/tex]
4. Interpret the discriminant: The value of the discriminant tells us about the number and 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 (a repeated root).
- If [tex]\( \Delta < 0 \)[/tex], there are no real solutions; instead, there are two complex conjugate solutions.
Since the discriminant is [tex]\( -4 \)[/tex], which is less than zero ([tex]\( \Delta < 0 \)[/tex]), the quadratic equation [tex]\( 0 = x^2 - 4x + 5 \)[/tex] has no real solutions, but it has two complex conjugate solutions.
Thus, the correct answer is:
The discriminant is [tex]\(-4\)[/tex], so the equation has no real solutions.
