What is the value of the discriminant, [tex]\( b^2 - 4ac \)[/tex], for the quadratic equation [tex]\( 0 = x^2 - 4x + 5 \)[/tex], and what does it mean about the number of real solutions the equation has?

A. The discriminant is -4, so the equation has no real solutions.
B. The discriminant is -4, so the equation has 2 real solutions.
C. The discriminant is 35, so the equation has 2 real solutions.
D. The discriminant is 35, so the equation has no real solutions.



Answer :

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.