To solve the quadratic equation [tex]\(x^2 - 8x + 41 = 0\)[/tex], we will follow the steps of completing the square and using the quadratic formula. The quadratic equation is generally solved in the form [tex]\(ax^2 + bx + c = 0\)[/tex].
Given equation:
[tex]\[x^2 - 8x + 41 = 0\][/tex]
Identify coefficients:
[tex]\[a = 1, \quad b = -8, \quad c = 41\][/tex]
The quadratic formula is given by:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
First, compute the discriminant ([tex]\(b^2 - 4ac\)[/tex]):
[tex]\[b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot 41\][/tex]
[tex]\[= 64 - 164\][/tex]
[tex]\[= -100\][/tex]
Since the discriminant is negative, the equation will have complex roots. Next, substitute the values into the quadratic formula:
[tex]\[x = \frac{-(-8) \pm \sqrt{-100}}{2 \cdot 1}\][/tex]
[tex]\[x = \frac{8 \pm \sqrt{-100}}{2}\][/tex]
[tex]\[x = \frac{8 \pm \sqrt{100}i}{2}\][/tex]
[tex]\[x = \frac{8 \pm 10i}{2}\][/tex]
Now, simplify the expression:
[tex]\[x = 4 \pm 5i\][/tex]
Thus, the solutions to the equation [tex]\(x^2 - 8x + 41 = 0\)[/tex] are:
[tex]\[x = 4 - 5i \quad \text{and} \quad x = 4 + 5i\][/tex]
