⚡ Toppers Hub: Class 10 Math Specimen Deck ⚡
Mastering High-Yield Board Exam Problems Step-by-Step!
📌 The Question
"Find two consecutive odd positive integers, the sum of whose squares is 290."
📋 Step-by-Step Answer
Let the first odd positive integer be x.
Since odd numbers have a gap of 2 (like 1, 3, 5, 7...), the next consecutive odd positive integer will be (x + 2).
According to the question, the sum of their squares is 290:
Expand the equation using the algebraic identity (a+b)² = a² + 2ab + b²:
2x² + 4x + 4 = 290
Bring 290 to the left side:
2x² + 4x - 286 = 0
Divide the entire equation by 2 to make it simpler:
We need two numbers that multiply to -143 and add up to 2. Those numbers are +13 and -11.
x(x + 13) - 11(x + 13) = 0
(x - 11)(x + 13) = 0
So, the possible values for x are:
x + 13 = 0 ⟹ x = -13
The question explicitly asks for positive integers. Therefore, we reject x = -13.
Now calculate both integers:
• The second consecutive odd integer (x + 2) = 11 + 2 = 13
🎉 Final Answer: 11 and 13 🎉
Self-Check Verification: 11² (121) + 13² (169) = 290. The logic holds perfectly!
💡 Topper's Board Presentation Secrets
In CBSE Board Evaluations, every stage carries individual step-marks. Missing simple components could cost you a top score. Always remember these guidelines:
1. The Rejection Reason
Whenever you obtain a negative value in real-world application questions (like dimensions, age, or specified positive integers), always explicitly mention a line explaining why you are ignoring it. Writing "Rejecting negative value since x must be a positive integer" guarantees full credit on the final step.
2. Alternative Solution Path
If splitting the middle term becomes confusing during exam hours, instantly shift to the quadratic formula tool to save time without stress:
- Identify your parameters clearly from the simplified expression: a = 1, b = 2, c = -143.
- Apply the absolute formula: x = [-b ± √(b² - 4ac)] / 2a.
- This method removes factoring guesswork completely when dealing with large numbers.