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Answer
The expression \( 6a^{2} - 36a + 48 \) can be factored as \( 6(a - 2)(a - 4) \).
Solution
Certainly! Let's factorize the quadratic expression:
\[ 6a^{2} - 36a + 48 \]
**Step 1: Factor Out the Greatest Common Factor (GCF)**
First, identify the greatest common factor of all the terms. The GCF of 6, 36, and 48 is **6**.
\[ 6a^{2} - 36a + 48 = 6(a^{2} - 6a + 8) \]
**Step 2: Factor the Quadratic Expression**
Now, focus on factoring the quadratic inside the parentheses:
\[ a^{2} - 6a + 8 \]
We need to find two numbers that multiply to **8** (the constant term) and add up to **-6** (the coefficient of the middle term).
Let's list the factor pairs of 8:
- 1 and 8
- 2 and 4
Now, considering the signs, since the product is positive and the sum is negative, both numbers should be negative:
- \(-2\) and \(-4\)
Check:
\[ (-2) + (-4) = -6 \]
\[ (-2) \times (-4) = 8 \]
**Step 3: Write the Factored Form**
Using these numbers, we can write the factored form of the quadratic:
\[ a^{2} - 6a + 8 = (a - 2)(a - 4) \]
**Step 4: Combine with the GCF**
Don't forget to include the GCF we factored out earlier:
\[ 6(a^{2} - 6a + 8) = 6(a - 2)(a - 4) \]
**Final Factored Form:**
\[ 6a^{2} - 36a + 48 = 6(a - 2)(a - 4) \]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To factor the expression \( 6a^2 - 36a + 48 \), first, look for a common factor. Here, 6 is a common factor for all terms, so we can factor it out: \( 6(a^2 - 6a + 8) \). Next, focus on factoring the quadratic \( a^2 - 6a + 8 \). We need two numbers that multiply to 8 and add up to -6, which are -2 and -4. Thus, \( a^2 - 6a + 8 \) factors to \( (a - 2)(a - 4) \). Putting it all together, the fully factored form of the expression is: \[ 6(a - 2)(a - 4) \] Now, doesn't that just feel like you’ve unlocked a little math magic? 🎩✨
