x²-2x+9=0

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:

\[ x = 1 + 2\sqrt{2}i \quad \text{and} \quad x = 1 - 2\sqrt{2}i \]