Completely factor the trinomial, if possible.

3x2−13x−103x2−13x−10




Let's factor the trinomial \( 3x^2 - 13x - 10 \).

### Step-by-step process:

1. **Identify the coefficients:**
   The trinomial is in the form \( ax^2 + bx + c \), where:
   - \( a = 3 \)
   - \( b = -13 \)
   - \( c = -10 \)

2. **Multiply \( a \) and \( c \):**
   Multiply the leading coefficient \( a \) and the constant \( c \):
   \[
   a \times c = 3 \times (-10) = -30
   \]

3. **Find two numbers that multiply to \( -30 \) and add to \( -13 \):**
   We need to find two numbers whose product is \( -30 \) and whose sum is \( -13 \). These numbers are \( -15 \) and \( 2 \) because:
   \[
   -15 \times 2 = -30 \quad \text{and} \quad -15 + 2 = -13
   \]

4. **Rewrite the middle term using these numbers:**
   Split the middle term \( -13x \) into \( -15x \) and \( 2x \):
   \[
   3x^2 - 15x + 2x - 10
   \]

5. **Factor by grouping:**
   Group the terms:
   \[
   (3x^2 - 15x) + (2x - 10)
   \]
   Factor each group:
   \[
   3x(x - 5) + 2(x - 5)
   \]

6. **Factor out the common binomial factor:**
   Both groups contain the common factor \( (x - 5) \), so factor it out:
   \[
   (3x + 2)(x - 5)
   \]

### Final factored form:
\[
(3x + 2)(x - 5)
\]

Would you like more details or have any questions?

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**Related questions:**
1. How do you factor trinomials with larger coefficients?
2. Can all trinomials be factored?
3. What is the significance of the discriminant in factoring?
4. How do you solve quadratic equations by factoring?
5. How can we verify the factored form of a quadratic equation?

**Tip:** When factoring trinomials, always check if the trinomial can be simplified or if any common factors exist before starting the process.

Completely factor the trinomial, if possible. 3x^2 - 13x - 10

Learn how to factor the trinomial 3x^2 - 13x - 10 using step-by-step algebraic techniques. This problem explains factoring by grouping, identifying coefficients, and finding the product-sum pair. Suitable for high school students in Grades 9-11.

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