Find two consecutive positive integers, sum of whose squares is 365.
The two consecutive positive integers are 13 and 14.
Explanation
This problem from Exercise 4.2 requires forming a quadratic equation from a word problem. The student must define consecutive integers as x and x+1, form the equation x² + (x+1)² = 365, and solve by factorisation. The context shows similar problem-solving approaches in Example 6 and Question 3, where quadratic equations are formed and solved using factorisation method.
Solution Steps
Step 1: Let the two consecutive positive integers be x and (x + 1).
Step 2: According to the question, sum of their squares = 365. So, x² + (x + 1)² = 365
Step 3: Expanding: x² + x² + 2x + 1 = 365, which gives 2x² + 2x – 364 = 0
Step 4: Dividing by 2: x² + x – 182 = 0
Step 5: Factorising: x² + 14x – 13x – 182 = 0, i.e., x(x + 14) – 13(x + 14) = 0
Step 6: So, (x – 13)(x + 14) = 0, giving x = 13 or x = –14
Step 7: Since we need positive integers, x = 13. Therefore, the integers are 13 and 14.