NECO 2002 Mathematics · Question 14
What must be added to 3x2 - 5x to make it a perfect square?
Options
- A.
- B.
- C.-
- D.-
- E.-
Answer: B.
Explanation written with AI and checked against the answer key
Analyze the question: The expression is probably 3x² - 5x (the "2" is the square sign). We need a constant to add so that it becomes a perfect square. Concept: Completing the square. For ax² + bx, factor out the coefficient of x² first, then add the square of half the coefficient of x inside the bracket. Solution: 3x² - 5x = 3(x² - (5/3)x) Half of 5/3 is 5/6, and its square is 25/36. So inside the bracket we add 25/36: 3(x² - (5/3)x + 25/36) = 3(x - 5/6)² The actual amount added to the original expression is 3 × 25/36 = 25/12. Check: 3x² - 5x + 25/12 = 3(x² - (5/3)x + 25/36) = 3(x - 5/6)². ✓. Comparing with the options: Strictly, the constant to add to 3x² - 5x is 25/12, which is not among the options. The option 25/36 is the number added inside the bracket after factoring out 3 (the usual "completing the square" step). Note that if you add 25/36 directly to 3x² - 5x, the result is not a perfect square: its discriminant is 25 - 4(3)(25/36) = 25 - 25/3 ≠ 0. So my working disagrees slightly with the literal wording, but B is the only option matching the standard completing-the-square step, and it is the key's answer. Tip: In completing the square, always make the coefficient of x² equal to 1 first, then take (half the coefficient of x)². In exams, if your exact answer isn't listed, check whether the question expects the value inside the bracket.AI can make mistakes. Report anything that looks wrong.
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