2. \( 3 x^{2}-6 x+6 \)

Certainly! It looks like you're working with the quadratic expression: \[ 3x^{2} - 6x + 6 \] Let's explore different ways to analyze and manipulate this expression. ### 1. **Factoring Out the Greatest Common Factor (GCF)** First, identify the greatest common factor of the coefficients: - The GCF of \(3\), \(-6\), and \(6\) is **3**. Factor out the GCF: \[ 3x^{2} - 6x + 6 = 3(x^{2} - 2x + 2) \] ### 2. **Determining if the Quadratic Can Be Factored Further** Next, check if the quadratic inside the parentheses can be factored over the real numbers. Consider the quadratic equation: \[ x^{2} - 2x + 2 \] To determine if it can be factored, calculate the **discriminant** (\(D\)): \[ D = b^{2} - 4ac = (-2)^2 - 4(1)(2) = 4 - 8 = -4 \] Since the discriminant is **negative** (\(D = -4\)), the quadratic does not have real roots and cannot be factored further using real numbers. However, it can be factored using complex numbers: \[ x = \frac{2 \pm \sqrt{-4}}{2} = 1 \pm i \] So, the factored form over the complex numbers is: \[ 3(x - (1 + i))(x - (1 - i)) \] ### 3. **Completing the Square** Another way to rewrite the quadratic is by completing the square: \[ 3x^{2} - 6x + 6 = 3(x^{2} - 2x) + 6 \] Complete the square inside the parentheses: \[ x^{2} - 2x = (x - 1)^2 - 1 \] Substitute back: \[ 3((x - 1)^2 - 1) + 6 = 3(x - 1)^2 - 3 + 6 = 3(x - 1)^2 + 3 \] So, the expression in completed square form is: \[ 3(x - 1)^2 + 3 \] ### 4. **Vertex Form and Graphing** From the completed square form \(3(x - 1)^2 + 3\), you can identify the vertex of the parabola represented by the quadratic function \(y = 3x^{2} - 6x + 6\): - **Vertex:** \((1, 3)\) - **Direction:** Since the coefficient of \((x - 1)^2\) is positive (\(3 > 0\)), the parabola opens **upwards**. ### 5. **Summary** - **Original Expression:** \(3x^{2} - 6x + 6\) - **Factored Form (over complex numbers):** \(3(x - (1 + i))(x - (1 - i))\) - **Completed Square Form:** \(3(x - 1)^2 + 3\) - **Vertex:** \((1, 3)\) If you have a specific operation or analysis you'd like to perform on this expression, feel free to let me know!