Solved the equation $2x^2 - x - 6$ = 0

Question analysis: Solve the quadratic equation 2x² − x − 6 = 0 by factorisation.

Concept: For ax² + bx + c = 0, find two numbers that multiply to a × c and add to b.

Solution:

• a × c = 2 × (−6) = −12
• b = −1
• The two numbers are −4 and 3, because (−4)(3) = −12 and −4 + 3 = −1.

Split the middle term:

2x² − 4x + 3x − 6 = 0

Group:

2x(x − 2) + 3(x − 2) = 0

(x − 2)(2x + 3) = 0

So:

• x − 2 = 0 gives x = 2
• 2x + 3 = 0 gives x = −3/2

Check:

• x = 2: 2(4) − 2 − 6 = 0 ✓
• x = −3/2: 2(9/4) + 3/2 − 6 = 4.5 + 1.5 − 6 = 0 ✓

Answer: x = −3/2 or 2, which is option A.

Tip: You can quickly test the options by substitution. Option B's x = 3/2 gives 2(2.25) − 1.5 − 6 = −3, which is not zero, so B is wrong. Also, the product of the roots should be c/a = −3, and (−3/2)(2) = −3 ✓. The sum of the roots should be −b/a = 1/2, and −3/2 + 2 = 1/2 ✓.