Question

look at the expression below and select the best answer.
$x^2 - 4x + 13$
○ no solution
○ complex solution(s)
○ imaginary solution(s)
○ real solution(s)

Explanation:

Step1: Recall the discriminant formula

For a quadratic equation \(ax^2 + bx + c = 0\), the discriminant is \(D = b^2 - 4ac\).
For the expression \(x^2 - 4x + 13\) (treating it as \(x^2 - 4x + 13 = 0\) for finding solutions), \(a = 1\), \(b = -4\), \(c = 13\).

Step2: Calculate the discriminant

$$
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$$

Step3: Analyze the discriminant

Since the discriminant \(D = -36<0\), the quadratic equation has complex solutions (complex numbers include real and imaginary parts; when discriminant is negative, solutions are of the form \(a + bi\) where \(b
eq0\), so they are complex solutions, and also can be called imaginary solutions in a sense, but "complex solutions" is a more general term here. However, let's check the options: "Complex Solution(s)" is a valid option as complex numbers include imaginary numbers. Wait, but also, when discriminant is negative, the solutions are complex (non - real) numbers. Let's re - check:

The discriminant is negative, so the solutions are not real. The solutions will be complex numbers (of the form \(2\pm3i\) when we solve \(x^2 - 4x + 13 = 0\) by completing the square: \(x^2 - 4x=-13\), \((x - 2)^2=-13 + 4=-9\), \(x - 2=\pm3i\), \(x = 2\pm3i\)). So the solutions are complex (and also imaginary in the sense of having an imaginary part). But among the options, "Complex Solution(s)" is a correct description as complex numbers include imaginary numbers. Wait, but let's see the options:

• "No Solution": Incorrect, because the equation \(x^2 - 4x + 13 = 0\) has solutions (complex ones).
• "Complex Solution(s)": Correct, because the solutions are complex numbers (\(2\pm3i\) are complex numbers).
• "Imaginary Solution(s)": Also correct in a way, but complex numbers are a broader category. However, let's check the discriminant: since \(D=-36\), the solutions are \(x=\frac{4\pm\sqrt{-36}}{2}=\frac{4\pm6i}{2}=2\pm3i\), which are complex numbers (and also imaginary - part non - zero). But the option "Complex Solution(s)" is appropriate. Also, note that sometimes "imaginary solutions" is used when the real part is zero, but here real part is 2, so they are complex (with real and imaginary parts). So the best answer among the options is "Complex Solution(s)". Wait, but also, "Imaginary Solution(s)" could be considered, but let's check the discriminant interpretation:

When discriminant is negative, the quadratic has two complex conjugate solutions (which are also called imaginary solutions in some contexts, but complex solutions is a more accurate term as they have both real and imaginary parts). So the correct option is "Complex Solution(s)".

Answer:

Complex Solution(s)