Question:
If the sum of the squares of two consecutive positive odd integers is 1354, then one of them is:
If the sum of the squares of two consecutive positive odd integers is 1354, then one of them is:
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To solve \( x^2 + 2x - 675 = 0 \) quickly, recognize that \( x^2 + 2x \approx x^2 \). Since \( \sqrt{675} \) is between \( 25 \) (\( 625 \)) and \( 26 \) (\( 676 \)), \( x = 25 \) is the most likely candidate.
Updated On: Jul 18, 2026
- a perfect square and the other is a perfect cube
- not a perfect square and the other is a perfect cube
- a perfect square and the other is not a perfect cube
- not a perfect square and the other is not a perfect cube
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The Correct Option is A
Solution and Explanation
Concept:
Consecutive odd integers can be represented as \( x \) and \( x+2 \), where \( x \) is an odd integer.
• Equation: \( x^2 + (x+2)^2 = 1354 \).
• Solve the quadratic equation for the positive integer \( x \).
Step 1: Setting up and solving the equation.
\[ x^2 + (x^2 + 4x + 4) = 1354 \] \[ 2x^2 + 4x - 1350 = 0 \quad \Rightarrow \quad x^2 + 2x - 675 = 0 \] Factoring the quadratic: \[ x^2 + 27x - 25x - 675 = 0 \quad \Rightarrow \quad x(x+27) - 25(x+27) = 0 \] \[ (x-25)(x+27) = 0 \] Since the integers are positive, \( x = 25 \).
Step 2: Identifying the integers.
The two consecutive odd integers are \( 25 \) and \( 25+2 = 27 \).
Step 3: Analyzing the properties of the integers.
- \( 25 = 5^2 \) (a perfect square). - \( 27 = 3^3 \) (a perfect cube). Thus, one is a perfect square and the other is a perfect cube.
• Equation: \( x^2 + (x+2)^2 = 1354 \).
• Solve the quadratic equation for the positive integer \( x \).
Step 1: Setting up and solving the equation.
\[ x^2 + (x^2 + 4x + 4) = 1354 \] \[ 2x^2 + 4x - 1350 = 0 \quad \Rightarrow \quad x^2 + 2x - 675 = 0 \] Factoring the quadratic: \[ x^2 + 27x - 25x - 675 = 0 \quad \Rightarrow \quad x(x+27) - 25(x+27) = 0 \] \[ (x-25)(x+27) = 0 \] Since the integers are positive, \( x = 25 \).
Step 2: Identifying the integers.
The two consecutive odd integers are \( 25 \) and \( 25+2 = 27 \).
Step 3: Analyzing the properties of the integers.
- \( 25 = 5^2 \) (a perfect square). - \( 27 = 3^3 \) (a perfect cube). Thus, one is a perfect square and the other is a perfect cube.
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