Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 6v^{2} - 5v - 6 = (3v + 2)(2v - 3) \)
Solution
To factor the quadratic expression \( 6v^{2} - 5v - 6 \) by grouping, follow these steps:
1. **Multiply the coefficient of \( v^2 \) by the constant term:**
\( 6 \times (-6) = -36 \)
2. **Find two numbers that multiply to \(-36\) and add up to \(-5\):**
The numbers are \(-9\) and \(4\), since:
\[
-9 \times 4 = -36 \quad \text{and} \quad -9 + 4 = -5
\]
3. **Rewrite the middle term using these numbers:**
\[
6v^{2} - 9v + 4v - 6
\]
4. **Factor by grouping:**
\[
(6v^{2} - 9v) + (4v - 6) = 3v(2v - 3) + 2(2v - 3)
\]
5. **Factor out the common binomial factor \((2v - 3)\):**
\[
(3v + 2)(2v - 3)
\]
**Final Factored Form:**
\[
(3v + 2)(2v - 3)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To factor the expression \( 6v^2 - 5v - 6 \) by grouping, we start by finding two numbers that multiply to \( 6 \cdot (-6) = -36 \) and add to \( -5 \). The numbers \( -9 \) and \( 4 \) fit the bill! Now, we can rewrite the expression: \[ 6v^2 - 9v + 4v - 6 \] Next, group the terms: \[ (6v^2 - 9v) + (4v - 6) \] Now, factor out the common factors in each group: \[ 3v(2v - 3) + 2(2v - 3) \] Now, you can see that \( (2v - 3) \) is a common factor: \[ (2v - 3)(3v + 2) \] So, the factored form of \( 6v^2 - 5v - 6 \) is: \[ (2v - 3)(3v + 2) \]
