To solve the quadratic equation [tex]\(0 = x^2 - 4x + 5\)[/tex] and determine the value of the discriminant as well as the number of real solutions, follow these steps:
1. Identify the coefficients: In the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], coefficients are:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = -4 \][/tex]
[tex]\[ c = 5 \][/tex]
2. Calculate the discriminant: The discriminant [tex]\(D\)[/tex] of a quadratic equation is given by:
[tex]\[ D = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[ D = (-4)^2 - 4 \cdot 1 \cdot 5 \][/tex]
3. Perform the operations:
[tex]\[ (-4)^2 = 16 \][/tex]
[tex]\[ 4 \cdot 1 \cdot 5 = 20 \][/tex]
[tex]\[ D = 16 - 20 \][/tex]
[tex]\[ D = -4 \][/tex]
4. Interpret the discriminant:
- If [tex]\(D > 0\)[/tex], the quadratic equation has 2 distinct real solutions.
- If [tex]\(D = 0\)[/tex], the quadratic equation has exactly 1 real solution.
- If [tex]\(D < 0\)[/tex], the quadratic equation has no real solutions.
Since the discriminant [tex]\(D = -4\)[/tex], which is less than zero, this means the quadratic equation [tex]\(0 = x^2 - 4x + 5\)[/tex] has no real solutions.
Therefore, the value of the discriminant is [tex]\(-4\)[/tex], and the correct interpretation is:
The discriminant is -4, so the equation has no real solutions.
