Solve 6x$^2$ = 5x - 1

Analyze the question: We need to find the values of x that satisfy the quadratic equation 6x² = 5x − 1.

Concept: Move every term to one side so the equation equals zero (ax² + bx + c = 0), then factorise.

Solution:

1. Rearrange: 6x² − 5x + 1 = 0

2. Factorise by finding two numbers that multiply to 6 × 1 = 6 and add to −5. These are −2 and −3.

3. Split the middle term: 6x² − 2x − 3x + 1 = 0

4. Group: 2x(3x − 1) − 1(3x − 1) = 0

5. Factor out the common bracket: (3x − 1)(2x − 1) = 0

6. Set each factor to zero: • 3x − 1 = 0 → x = 1/3 • 2x − 1 = 0 → x = 1/2

Check:

• x = 1/2: 6(1/4) = 1.5 and 5(1/2) − 1 = 1.5 ✓
• x = 1/3: 6(1/9) = 2/3 and 5(1/3) − 1 = 2/3 ✓

Answer: C

Tip: After rearranging, you can quickly test the options by substituting them back. Also, if the middle term is negative and the last term is positive (like −5x and +1), both roots will be positive. That immediately rules out option D, which has a negative root. For options A and B, the values 2, 3, and 0 are too large or don't work, so you can eliminate them fast.