It looks like you have the quadratic expression:

\[ 2a^{2} - 24a + 70 \]

Let's factor this expression step by step.

### Step 1: Factor Out the Greatest Common Factor (GCF)
First, identify and factor out the GCF from all the terms. Here, the GCF is **2**.

\[
2a^{2} - 24a + 70 = 2(a^{2} - 12a + 35)
\]

### Step 2: Factor the Quadratic Inside the Parentheses
Now, focus on factoring the quadratic expression inside the parentheses:

\[
a^{2} - 12a + 35
\]

We need to find two numbers that:
- **Multiply to give +35**
- **Add up to -12**

The numbers **-5** and **-7** satisfy these conditions because:

\[
(-5) \times (-7) = 35 \quad \text{and} \quad (-5) + (-7) = -12
\]

### Step 3: Write the Factored Form
Using these numbers, the quadratic can be factored as:

\[
a^{2} - 12a + 35 = (a - 5)(a - 7)
\]

### Final Factored Expression
Combine this with the GCF you factored out earlier:

\[
2(a^{2} - 12a + 35) = 2(a - 5)(a - 7)
\]

**So, the fully factored form of the original expression is:**

\[
2(a - 5)(a - 7)
\]

Home Question Bank Math Algebra

Question

c) \( 2 a^{2}-24 a+70 \)

Ask by Whittaker Harris. in South Africa

Feb 03,2025

Real Tutor Solution

Tutor-Verified Answer

Answer

The expression \( 2a^{2} - 24a + 70 \) can be factored as \( 2(a - 5)(a - 7) \).

Solution

It looks like you have the quadratic expression: \[ 2a^{2} - 24a + 70 \] Let's factor this expression step by step. ### Step 1: Factor Out the Greatest Common Factor (GCF) First, identify and factor out the GCF from all the terms. Here, the GCF is **2**. \[ 2a^{2} - 24a + 70 = 2(a^{2} - 12a + 35) \] ### Step 2: Factor the Quadratic Inside the Parentheses Now, focus on factoring the quadratic expression inside the parentheses: \[ a^{2} - 12a + 35 \] We need to find two numbers that: - **Multiply to give +35** - **Add up to -12** The numbers **-5** and **-7** satisfy these conditions because: \[ (-5) \times (-7) = 35 \quad \text{and} \quad (-5) + (-7) = -12 \] ### Step 3: Write the Factored Form Using these numbers, the quadratic can be factored as: \[ a^{2} - 12a + 35 = (a - 5)(a - 7) \] ### Final Factored Expression Combine this with the GCF you factored out earlier: \[ 2(a^{2} - 12a + 35) = 2(a - 5)(a - 7) \] **So, the fully factored form of the original expression is:** \[ 2(a - 5)(a - 7) \]

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The Deep Dive

To factor the expression \( 2a^{2} - 24a + 70 \), we can first factor out the greatest common factor, which is 2. This simplifies our expression to \( 2(a^{2} - 12a + 35) \). Now, we look to factor the quadratic \( a^{2} - 12a + 35 \), which can be factored as \( (a - 5)(a - 7) \). So, the fully factored form of the original expression is \( 2(a - 5)(a - 7) \). Did you know that factoring quadratics is like solving a puzzle? It’s all about finding two numbers that multiply to the constant term while adding up to the linear coefficient. Mistakes often happen when we overlook the signs of the numbers or forget to double-check our multiplication. Always remember to use the FOIL method to verify your factors! If you're keen to dive deeper into quadratic equations and their applications, you might enjoy exploring algebra textbooks that provide a wealth of examples and practice problems. Alternatively, websites like Khan Academy offer interactive lessons and quizzes that can solidify your understanding while keeping the learning process light and enjoyable!

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