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Answer
The quadratic expression \(2a^{2} - 24a + 20\) can be factored as \(2(a - (6 + \sqrt{26}))(a - (6 - \sqrt{26}))\).
Solution
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!
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To start, this expression can be factored or simplified! First, we notice that we can factor out a common term from all parts of the polynomial. Let’s take out a 2 for an easier factorization: \[ 2(a^2 - 12a + 10) \] Now, we can work on the quadratic \( a^2 - 12a + 10 \). To factor this further, we can use the quadratic formula or complete the square approach. Also, this expression can easily pop up in real-world scenarios like optimizing profits or costs in a business model. By tweaking the values of \(a\), businesses can find the most efficient level of production! The world of algebra isn’t just about numbers; it’s about finding solutions to practical problems too!
