Question
(1) Factor completely: \( 6 x+24 \) (2) Factor completely: \( 6 x^{2}+11 x-10 \)
Ask by Rodriquez Mullins. in the United States
Feb 02,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(6(x + 4)\) and \((3x - 2)(2x + 5)\).
Solution
Let's factor each expression step by step.
### (1) Factor \( 6x + 24 \)
**Step 1:** Identify the greatest common factor (GCF) in the terms \(6x\) and \(24\).
- The GCF of 6 and 24 is **6**.
**Step 2:** Factor out the GCF.
\[
6x + 24 = 6(x) + 6(4) = 6(x + 4)
\]
**Answer:** \( 6(x + 4) \)
---
### (2) Factor \( 6x^2 + 11x - 10 \)
**Step 1:** Look for two numbers that multiply to \(6 \times (-10) = -60\) and add up to **11**.
- The numbers **15** and **-4** satisfy this: \(15 \times (-4) = -60\) and \(15 + (-4) = 11\).
**Step 2:** Rewrite the middle term using these numbers.
\[
6x^2 + 15x - 4x - 10
\]
**Step 3:** Factor by grouping.
\[
= (6x^2 + 15x) + (-4x - 10) \\
= 3x(2x + 5) - 2(2x + 5) \\
= (3x - 2)(2x + 5)
\]
**Answer:** \( (3x - 2)(2x + 5) \)
---
**Final Answers:**
1. \(6(x + 4)\)
2. \((3x - 2)(2x + 5)\)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
Let's dive into the factoring fun! To tackle \( 6x + 24 \), first, let's spot the common factor. Both terms have a 6 in common, so we factor that out, giving us \( 6(x + 4) \). Voila! We've transformed it into a simpler expression by spotting that common factor! Now, for \( 6x^2 + 11x - 10 \), we can use the "ac method" to factor this quadratic expression. First, multiply \( a \) and \( c\) (6 and -10), which gives us -60. We need two numbers that multiply to -60 and add to 11. Those numbers are 15 and -4. Rewrite the expression as \( 6x^2 + 15x - 4x - 10 \), then group terms: \( 3x(2x + 5) - 2(2x + 5) \). Factoring by grouping, we get \( (2x + 5)(3x - 2) \). Super cool, right?
