The product of two consecutive odd integers is 143. What are the integers? - Decision Point
The Product of Two Consecutive Odd Integers Is 143 – Find 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 Context
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
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Key Insights
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
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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:
