x^2-6x+3=0

To solve the quadratic equation \(x^2 - 6x + 3 = 0\), you can use the quadratic formula:

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

In this equation, the coefficients are:

• \(a = 1\)
• \(b = -6\)
• \(c = 3\)

Now, plug these values into the quadratic formula:

1. Calculate the discriminant \(b^2 - 4ac\): \[ b^2 - 4ac = (-6)^2 - 4(1)(3) = 36 - 12 = 24 \]

2. Substitute the values into the formula: \[ x = \frac{-(-6) \pm \sqrt{24}}{2(1)} = \frac{6 \pm \sqrt{24}}{2} \]

3. Simplify \(\sqrt{24}\): \[ \sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6} \]

4. Substitute this back into the equation: \[ x = \frac{6 \pm 2\sqrt{6}}{2} \]

5. Simplify further: \[ x = 3 \pm \sqrt{6} \]

Thus, the solutions to the equation \(x^2 - 6x + 3 = 0\) are: \[ x = 3 + \sqrt{6} \quad \text{and} \quad x = 3 - \sqrt{6} \]