Factorise: $$ 3 x^{2}-13 x-10 $$

Question

UpStudy AI · Accepted Answer

To factorize the quadratic expression $$ 3x^{2} - 13x - 10 $$, we'll follow these steps:

1. **Identify the coefficients:**
   - $$ a = 3 $$
   - $$ b = -13 $$
   - $$ c = -10 $$

2. **Find two numbers that multiply to $$ a \cdot c = 3 \cdot (-10) = -30 $$** and add up to $$ b = -13 $$.

The numbers that satisfy these conditions are **-15** and **2** because:
   - $$ -15 \times 2 = -30 $$
   - $$ -15 + 2 = -13 $$

3. **Rewrite the middle term using these numbers:**
   $$
   3x^{2} - 15x + 2x - 10
   $$

4. **Factor by grouping:**
   $$
   (3x^{2} - 15x) + (2x - 10) \
   = 3x(x - 5) + 2(x - 5)
   $$

5. **Factor out the common binomial factor $$(x - 5)$$:**
   $$
   (3x + 2)(x - 5)
   $$

**Final Factorization:**
$$
3x^{2} - 13x - 10 = (3x + 2)(x - 5)
$$

**Answer:** 
$$
3x^{2} -13x -10 = (3x + 2)\, (x - 5)
$$
$$ 3x^{2} -13x -10 = (3x + 2)\, (x - 5) $$

Factorise: \( 3 x^{2}-13 x-10 \)