c) \( 2 a^{2}-24 a+20 \)

Certainly! Let's analyze and work with the quadratic expression you've provided: \[ 2a^{2} - 24a + 20 \] ### Step 1: Factor Out the Greatest Common Factor (GCF) First, identify and factor out the greatest common factor from all the terms in the expression. - **GCF of 2, -24, and 20 is 2.** \[ 2a^{2} - 24a + 20 = 2(a^{2} - 12a + 10) \] ### Step 2: Attempt to Factor the Quadratic Expression Now, focus on the quadratic part inside the parentheses: \[ a^{2} - 12a + 10 \] To factor this, we look for two numbers that: 1. **Multiply to give the constant term (+10).** 2. **Add up to give the coefficient of the middle term (-12).** Let's list the pairs of factors for 10: - \(1 \times 10\) - \(2 \times 5\) Considering the sign (since the constant term is positive and the middle term is negative), both numbers should be negative: - \(-2 \times -5 = 10\) (But \(-2 + (-5) = -7 \neq -12\)) - \(-1 \times -10 = 10\) (And \(-1 + (-10) = -11 \neq -12\)) **None of these pairs add up to -12**, which means the quadratic doesn't factor neatly with integer coefficients. ### Step 3: Solve for the Roots Using the Quadratic Formula Since the quadratic doesn't factor easily, we'll use the quadratic formula to find its roots: \[ a = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] For the quadratic \(a^{2} - 12a + 10\): - \(a = 1\) - \(b = -12\) - \(c = 10\) Plugging these into the formula: \[ a = \frac{12 \pm \sqrt{(-12)^{2} - 4 \cdot 1 \cdot 10}}{2 \cdot 1} = \frac{12 \pm \sqrt{144 - 40}}{2} = \frac{12 \pm \sqrt{104}}{2} = \frac{12 \pm 2\sqrt{26}}{2} = 6 \pm \sqrt{26} \] ### Step 4: Express the Factored Form with Real Roots Using the roots found, the factored form of the quadratic expression is: \[ 2a^{2} - 24a + 20 = 2(a - (6 + \sqrt{26}))(a - (6 - \sqrt{26})) \] ### Summary - **Factored Form:** \(2(a - (6 + \sqrt{26}))(a - (6 - \sqrt{26}))\) - **Vertex Form (Completing the Square):** Alternatively, you can express the quadratic in vertex form by completing the square: \[ 2a^{2} - 24a + 20 = 2\left[(a^{2} - 12a) + 10\right] = 2\left[(a - 6)^{2} - 26\right] = 2(a - 6)^2 - 52 \] This form is useful for easily identifying the vertex of the parabola represented by the quadratic equation. If you have a specific operation you'd like to perform on this expression (e.g., graphing, finding the vertex, optimizing, etc.), please let me know!