The product of two consecutive odd integers is 143. What are the integers? - SUpost
The Product of Two Consecutive Odd Integers Is 143 β Find the Integers
The Product of Two Consecutive Odd Integers Is 143 β 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, weβll 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. Letβs 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 Γ (x + 2) = 143
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Key Insights
Setting Up the Equation
Expand the equation:
xΒ² + 2x = 143
Bring all terms to one side to form a quadratic equation:
xΒ² + 2x β 143 = 0
Final Thoughts
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 β143
- Add to 2 (the coefficient of x)
Factoring 143:
143 = 11 Γ 13
So, β11 and +13 multiply to β143 and add to 2 β
Thus, factor the equation:
(x + 11)(x β 13) = 0
Wait β actually, (x + 11)(x β 13) = xΒ² β 2x β 143 β not our equation. We need (x + 11)(x β 13) = xΒ² β 2x β 143, but our equation is xΒ² + 2x β 143.
Letβs correct: we want two numbers that multiply to β143 and add to +2. Try:
11 and β13? β no, add to β2
Try β11 and 13? β add to 2 β yes! But signs differ.
Actually, correct factoring candidates:
