Question

solve for \\(x\\) in the equation \\(x^2 - 8x + 41 = 0\\)

\\(x = -4 \pm \sqrt{37}i\\)
\\(x = -4 \pm 5i\\)
\\(x = 4 \pm \sqrt{37}i\\)
\\(x = 4 \pm 5i\\)

Explanation:

Identify coefficients of the quadratic equation

$$ a = 1, \quad b = -8, \quad c = 41 $$

Apply the quadratic formula

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

$$ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(41)}}{2(1)} $$

Simplify the expression

$$ x = \frac{8 \pm \sqrt{64 - 164}}{2} $$

$$ x = \frac{8 \pm \sqrt{-100}}{2} $$

$$ x = \frac{8 \pm 10i}{2} $$

$$ x = 4 \pm 5i $$

Answer:

• (A) \(x = -4 \pm \sqrt{37}i\)
• (B) \(x = -4 \pm 5i\)
• (C) \(x = 4 \pm \sqrt{37}i\)
• (D) \(x = 4 \pm 5i\) (Correct answer)