To solve the quadratic equation [tex]\( x^2 - 8x + 41 = 0 \)[/tex], we follow these steps:
### Step 1: Identify the coefficients
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. Here, we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 41 \)[/tex]
### Step 2: Calculate the discriminant
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]
Substitute the given values:
[tex]\[ \Delta = (-8)^2 - 4 \cdot 1 \cdot 41 \][/tex]
[tex]\[ \Delta = 64 - 164 \][/tex]
[tex]\[ \Delta = -100 \][/tex]
### Step 3: Determine the nature of the roots
Since the discriminant is negative ([tex]\( \Delta < 0 \)[/tex]), the quadratic equation has two complex roots.
### Step 4: Find the roots using the quadratic formula
The quadratic formula for the roots of the equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex]:
[tex]\[ x = \frac{-(-8) \pm \sqrt{-100}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{8 \pm \sqrt{-100}}{2} \][/tex]
### Step 5: Simplify the expression
Recall that [tex]\( \sqrt{-100} = \sqrt{100 \cdot (-1)} = 10i \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit. So:
[tex]\[ x = \frac{8 \pm 10i}{2} \][/tex]
[tex]\[ x = 4 \pm 5i \][/tex]
### Conclusion
The roots of the quadratic equation [tex]\( x^2 - 8x + 41 = 0 \)[/tex] are [tex]\( x = 4 \pm 5i \)[/tex].
Thus, the correct answer is:
[tex]\[ x = 4 \pm 5i \][/tex]
