Find the values of k in the equation 6k2 = 5k + 6

Question analysis: We need to solve the quadratic equation 6k² = 5k + 6 (the "2" is the square: k²).

Concept: Move every term to one side so the equation equals zero, then factorise (or use the quadratic formula).

Solution:

1. Rearrange: 6k² − 5k − 6 = 0

2. Find two numbers that multiply to 6 × (−6) = −36 and add to −5. These are −9 and +4.

3. Split the middle term: 6k² − 9k + 4k − 6 = 0

4. Group and factorise: 3k(2k − 3) + 2(2k − 3) = 0 (2k − 3)(3k + 2) = 0

5. Solve each factor: • 2k − 3 = 0 gives k = 3/2 • 3k + 2 = 0 gives k = −2/3

Check: For k = 3/2: 6(9/4) = 13.5, and 5(3/2) + 6 = 7.5 + 6 = 13.5 ✓ For k = −2/3: 6(4/9) = 8/3, and 5(−2/3) + 6 = −10/3 + 18/3 = 8/3 ✓

So k = −2/3 or k = 3/2, which is option B.

Tip: Always rewrite the equation as ax² + bx + c = 0 first. To factorise, look for two numbers that multiply to a × c and add to b. You can then check quickly by substituting your answers back into the original equation.