The product of two consecutive odd integers is 143. What are the integers?
The Product of Two Consecutive Odd Integers Is 143 – Find the Integers
Have you ever wondered how to solve a simple yet intriguing math puzzle? One classic example is finding two consecutive odd integers whose product equals 143. In this article, we’ll explore how to identify these integers step by step and understand the logic behind their connection to the number 143.
Understanding the Problem
We are told that the product of two consecutive odd integers is 143. Let’s define these integers algebraically:
Let the first odd integer be x Then the next consecutive odd integer is x + 2 (since odd numbers are two units apart).
Thus, we write the equation: x × (x + 2) = 143
Setting Up the Equation
Expand the equation: x² + 2x = 143
Bring all terms to one side to form a quadratic equation: x² + 2x – 143 = 0
Solving the Quadratic Equation
We can solve this using factoring, completing the square, or the quadratic formula. Let's attempt factoring.
We need two numbers that:
- Multiply to –143
- Add to 2 (the coefficient of x)
Factoring 143: 143 = 11 × 13 So, –11 and +13 multiply to –143 and add to 2 ✅
Thus, factor the equation: (x + 11)(x – 13) = 0
Wait — actually, (x + 11)(x – 13) = x² – 2x – 143 — not our equation. We need (x + 11)(x – 13) = x² – 2x – 143, but our equation is x² + 2x – 143.
Let’s correct: we want two numbers that multiply to –143 and add to +2. Try:
11 and –13? → no, add to –2 Try –11 and 13? → add to 2 → yes! But signs differ.
Actually, correct factoring candidates:
We want: (x + a)(x – b) = 0, with a – b = 2 and a × b = 143
Try a = 13, b = 11 → 13 × 11 = 143, and 13 – 11 = 2 → perfect!
But since the middle term is positive +2x, we need: (x – 11)(x + 13) = x² + 2x – 143 ✅
So the correct factorization is: (x – 11)(x + 13) = 0 — but this does not match the middle term structure.
Wait — better: since the factors multiply to –143 and add to +2, and we want two numbers differing by 2 in sum but opposite sign difference.
Standard method: use quadratic formula instead.
Using the Quadratic Formula
From: x² + 2x – 143 = 0 Using the quadratic formula: x = [–b ± √(b² – 4ac)] / (2a) Where a = 1, b = 2, c = –143
x = [–2 ± √(2² – 4(1)(–143))] / 2 x = [–2 ± √(4 + 572)] / 2 x = [–2 ± √576] / 2 √576 = 24
So: x = [–2 + 24]/2 = 22/2 = 11 or x = [–2 – 24]/2 = –26/2 = –13
Thus, the two consecutive odd integers are 11 and 13 (since 11 × 13 = 143), and they are consecutive odd numbers.
Why These Are Consecutive Odd Integers
11 is odd, 11 + 2 = 13 is the next odd integer — they are indeed consecutive odd integers.
Check: 11 × 13 = 143 ✔️ Product matches given condition.
Why This Matters – The Math Behind It
This problem demonstrates how quadratic equations naturally arise in real-world number puzzles. Recognizing that consecutive odd integers differ by 2 allows algebraic modeling, and solving quadratics helps find precise integer solutions. It’s a classic example of algebra in action, helpful for students and math enthusiasts alike.
Summary
- Set unknowns as x and x + 2, consecutive odd integers
- Formula equation: x(x + 2) = 143 → x² + 2x – 143 = 0
- Solve via factoring (11 and 13) or quadratic formula
- Solution: 11 and 13 satisfy both being odd and consecutive, with product 143
Final Answer: The two consecutive odd integers are 11 and 13.
Keywords: consecutive odd integers, product of two odd integers 143, solve x(x+2)=143, quadratic equation odd integers, integer solutions 143, math puzzle consecutive odds, odd integer product 143
Meta Description: Find the two consecutive odd integers whose product is 143. Solve the equation x(x+2)=143 and discover 11 and 13 are the solution.
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