Find the roots of the equations: $3m^2 - 2m - 65 = 0$

Analyze the question: We need the values of m that satisfy the quadratic 3m² − 2m − 65 = 0.

Concept: Factorise by splitting the middle term. Find two numbers whose product is a × c = 3 × (−65) = −195 and whose sum is −2.

Solution:

Factors of 195 that differ by 2: 13 and 15. So we use −15 and +13 (product −195, sum −2).

3m² − 15m + 13m − 65 = 0

3m(m − 5) + 13(m − 5) = 0

(m − 5)(3m + 13) = 0

So:

• m − 5 = 0 → m = 5
• 3m + 13 = 0 → m = −13/3

Check:

• m = 5: 3(25) − 10 − 65 = 75 − 75 = 0 ✓
• m = −13/3: 3(169/9) + 26/3 − 65 = 169/3 + 26/3 − 195/3 = 0 ✓

The roots are m = 5 and m = −13/3, which matches option B.

Tip: For quick checking, the sum of the roots should equal −b/a = 2/3. Here 5 + (−13/3) = 2/3 ✓. The product should equal c/a = −65/3. Here 5 × (−13/3) = −65/3 ✓. You can use this to eliminate wrong options fast in the exam.