To determine the number of real-number solutions for the quadratic equation [tex]\(x^2 - 4x + 6 = 0\)[/tex], we can use the discriminant method. The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Here, the coefficients are:
[tex]\[
a = 1, \quad b = -4, \quad c = 6
\][/tex]
Plugging these values into the formula for the discriminant:
[tex]\[
\Delta = (-4)^2 - 4(1)(6)
\][/tex]
[tex]\[
\Delta = 16 - 24
\][/tex]
[tex]\[
\Delta = -8
\][/tex]
The discriminant [tex]\(\Delta\)[/tex] determines the nature and number of the real-number solutions:
- If [tex]\(\Delta > 0\)[/tex], there are two distinct real-number solutions.
- If [tex]\(\Delta = 0\)[/tex], there is exactly one real-number solution.
- If [tex]\(\Delta < 0\)[/tex], there are no real-number solutions (the roots are complex or imaginary).
In this case, the discriminant is [tex]\(\Delta = -8\)[/tex], which is less than 0. Thus, the quadratic equation [tex]\(x^2 - 4x + 6 = 0\)[/tex] has no real-number solutions.
Therefore, the correct answer is:
C. The equation has no real-number solution.
