2023 · Q22
If (3a + 6b + c + 2d) × (3a − 6b − c + 2d) = (3a − 6b + c − 2d) × (3a + 6b − c − 2d), then which one of the following is correct?
Elementary MathematicsAlgebra
- Aab = cd
- Bac = bd
- Cad = bc
- Dad + bc = 0
Explanation
Rewrite each side as a difference of squares: LHS = [(3a+2d) + (6b+c)][(3a+2d) − (6b+c)] = (3a+2d)² − (6b+c)², and RHS = [(3a−2d) − (6b−c)][(3a−2d) + (6b−c)] = (3a−2d)² − (6b−c)². Expanding and cancelling the common terms 9a², 4d², −36b² and −c² leaves 12ad − 12bc = −12ad + 12bc on the two sides. Hence 24ad = 24bc, i.e. ad = bc. The answer is (C).