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The factored form of \( 6v^{2} - 6v - 36 \) is \( 6(v - 3)(v + 2) \).
Solución
To factor the quadratic expression \( 6v^{2} - 6v - 36 \), follow these steps:
1. **Factor Out the Greatest Common Factor (GCF):**
First, observe that each term in the expression shares a common factor of 6.
\[
6v^{2} - 6v - 36 = 6(v^{2} - v - 6)
\]
2. **Factor the Quadratic Inside the Parentheses:**
Now, focus on factoring the quadratic expression \( v^{2} - v - 6 \).
- **Find two numbers that multiply to \(-6\) and add to \(-1\):**
These numbers are \(-3\) and \(+2\) because:
\[
(-3) \times (+2) = -6 \quad \text{and} \quad (-3) + (+2) = -1
\]
- **Write the Factored Form:**
\[
v^{2} - v - 6 = (v - 3)(v + 2)
\]
3. **Combine the Factored Parts:**
Substitute the factored quadratic back into the expression:
\[
6(v^{2} - v - 6) = 6(v - 3)(v + 2)
\]
**Final Factored Form:**
\[
6v^{2} - 6v - 36 = 6(v - 3)(v + 2)
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Extra Insights
To solve the expression \( 6v^{2} - 6v - 36 \), you can factor out the common term, which is 6. This simplifies the equation to \( 6(v^{2} - v - 6) \). Next, you can factor \( v^{2} - v - 6 \) further into \( (v - 3)(v + 2) \). So, the complete factored form is \( 6(v - 3)(v + 2) \). Now you have a clear way to solve for \( v \), making it easier to find the values where the expression equals zero. If you set \( 6(v - 3)(v + 2) = 0 \), you can quickly determine that \( v = 3 \) and \( v = -2 \) are the roots of the equation. Happy factoring!
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