Answer :
To determine the discriminant of the quadratic equation [tex]\(0 = x^2 - 4x + 5\)[/tex] and what it means about the number of real solutions the equation has, we follow the steps below:
1. Identify the coefficients:
The quadratic equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex]. Comparing [tex]\(0 = x^2 - 4x + 5\)[/tex] with [tex]\(ax^2 + bx + c = 0\)[/tex], we get:
- [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]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we have:
[tex]\[ \Delta = (-4)^2 - 4 \cdot 1 \cdot 5 = 16 - 20 = -4 \][/tex]
So, the discriminant is [tex]\(-4\)[/tex].
3. Interpret the discriminant:
The value of the discriminant determines the number and nature of the solutions 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.
- If [tex]\(\Delta < 0\)[/tex], there are no real solutions (the solutions are complex).
In this case, since the discriminant is [tex]\(\Delta = -4\)[/tex], which is less than zero, the quadratic equation has no real solutions. Instead, the solutions are complex numbers.
Thus, the correct interpretation is:
The discriminant is -4, so the equation has no real solutions.
1. Identify the coefficients:
The quadratic equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex]. Comparing [tex]\(0 = x^2 - 4x + 5\)[/tex] with [tex]\(ax^2 + bx + c = 0\)[/tex], we get:
- [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]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we have:
[tex]\[ \Delta = (-4)^2 - 4 \cdot 1 \cdot 5 = 16 - 20 = -4 \][/tex]
So, the discriminant is [tex]\(-4\)[/tex].
3. Interpret the discriminant:
The value of the discriminant determines the number and nature of the solutions 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.
- If [tex]\(\Delta < 0\)[/tex], there are no real solutions (the solutions are complex).
In this case, since the discriminant is [tex]\(\Delta = -4\)[/tex], which is less than zero, the quadratic equation has no real solutions. Instead, the solutions are complex numbers.
Thus, the correct interpretation is:
The discriminant is -4, so the equation has no real solutions.
