To determine the nature of the solutions for the function [tex]\( f(x) = 2(x-1)^2 + 4 \)[/tex], we need to consider it in the form of a quadratic equation.
1. Identify the standard form of a quadratic equation:
Quadratic equations are typically written in the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
2. Express the function in standard form:
Since [tex]\( f(x) = 2(x-1)^2 + 4 \)[/tex], we can begin by expanding the squared term:
[tex]\[ f(x) = 2(x-1)^2 + 4 \][/tex]
[tex]\[ f(x) = 2(x^2 - 2x + 1) + 4 \][/tex]
[tex]\[ f(x) = 2x^2 - 4x + 2 + 4 \][/tex]
[tex]\[ f(x) = 2x^2 - 4x + 6 \][/tex]
3. Identify coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
From the expanded form [tex]\( 2x^2 - 4x + 6 \)[/tex], we can identify:
[tex]\[ a = 2 \][/tex]
[tex]\[ b = -4 \][/tex]
[tex]\[ c = 6 \][/tex]
4. Calculate the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we get:
[tex]\[ \Delta = (-4)^2 - 4(2)(6) \][/tex]
[tex]\[ \Delta = 16 - 48 \][/tex]
[tex]\[ \Delta = -32 \][/tex]
5. Analyze the discriminant:
The value of the discriminant [tex]\(\Delta\)[/tex] helps us determine the nature of the roots:
- If [tex]\(\Delta > 0\)[/tex], the quadratic equation has two distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], the quadratic equation has exactly one real root (a repeated root).
- If [tex]\(\Delta < 0\)[/tex], the quadratic equation has no real roots but two complex roots.
Since our discriminant [tex]\(\Delta = -32\)[/tex] is less than zero, the quadratic equation [tex]\(f(x) = 2x^2 - 4x + 6\)[/tex] has no real solutions but two complex solutions.
Therefore, the correct answer is:
D. two complex solutions
