To solve the quadratic equation ( x^2 - 2x + 9 = 0 ), we can use the quadratic formula:
[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]
In this equation, the coefficients are:
- ( a = 1 )
- ( b = -2 )
- ( c = 9 )
First, calculate the discriminant ( b^2 - 4ac ):
[
b^2 - 4ac = (-2)^2 - 4(1)(9) = 4 - 36 = -32
]
Since the discriminant is negative (( -32 < 0 )), this means there are no real solutions, but there are two complex solutions.
Now, we can proceed with the quadratic formula:
[
x = \frac{-(-2) \pm \sqrt{-32}}{2(1)} = \frac{2 \pm \sqrt{-32}}{2}
]
We can simplify ( \sqrt{-32} ):
[
\sqrt{-32} = \sqrt{32}i = \sqrt{16 \cdot 2}i = 4\sqrt{2}i
]
Now substituting this back into the formula:
[
x = \frac{2 \pm 4\sqrt{2}i}{2} = 1 \pm 2\sqrt{2}i
]
Therefore, the solutions to the equation ( x^2 - 2x + 9 = 0 ) are:
[
x = 1 + 2\sqrt{2}i \quad \text{and} \quad x = 1 - 2\sqrt{2}i
]
