Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 6g^{2} + 23g + 7 = (2g + 7)(3g + 1) \)
Solution
To factor the quadratic polynomial \( 6g^{2} + 23g + 7 \), follow these steps:
1. **Identify the coefficients:**
- \( a = 6 \)
- \( b = 23 \)
- \( c = 7 \)
2. **Multiply \( a \) and \( c \):**
\[
a \times c = 6 \times 7 = 42
\]
3. **Find two numbers that multiply to 42 and add up to 23:**
- The numbers are 2 and 21, since \( 2 \times 21 = 42 \) and \( 2 + 21 = 23 \).
4. **Rewrite the middle term using these numbers:**
\[
6g^2 + 2g + 21g + 7
\]
5. **Factor by grouping:**
\[
(6g^2 + 2g) + (21g + 7) = 2g(3g + 1) + 7(3g + 1)
\]
6. **Factor out the common binomial:**
\[
(2g + 7)(3g + 1)
\]
**Final Factored Form:**
\[
6g^{2} + 23g + 7 = (2g + 7)(3g + 1)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To factor the polynomial \( 6g^2 + 23g + 7 \), we can look for two numbers that multiply to \( 6 \times 7 = 42 \) and add up to \( 23 \). The numbers \( 21 \) and \( 2 \) fit this. Rewriting the middle term, we have: \[ 6g^2 + 21g + 2g + 7 \] Now, group the terms: \[ (6g^2 + 21g) + (2g + 7) \] Factor by grouping: \[ 3g(2g + 7) + 1(2g + 7) \] This gives us: \[ (3g + 1)(2g + 7) \] So, the factored form of the polynomial is \( (3g + 1)(2g + 7) \). If you're interested in checking your work, you can always expand the factors back out to make sure you arrive at the original polynomial again—it's a foolproof method! Plus, it’s satisfying to see how the pieces fit back together!
