Answer each question. Answer each question. Answer each question. Unknown NumbersUse the following facts.If x represents an integer, then x+1 represents the next consecutive integer.If x represents an even integer, then x+2 represents the next consecutive even integer.If x represents an odd integer, then x+2 represents the next consecutive odd integer. Find two consecutive odd integers whose product is 63.

Verified step by step guidance

1. Let the first odd integer be represented by \( x \).
2. The next consecutive odd integer can be represented as \( x + 2 \).
3. Set up the equation for their product: \( x(x + 2) = 63 \).
4. Expand the equation: \( x^2 + 2x = 63 \).
5. Rearrange the equation to form a quadratic equation: \( x^2 + 2x - 63 = 0 \).

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow one another in order without any gaps. For example, if x is an integer, then x and x+1 are consecutive integers. This concept is crucial for understanding how to identify pairs of integers that are next to each other, which is necessary for solving problems involving consecutive numbers.

Odd and Even Integers

Odd integers are numbers that cannot be divided evenly by 2, such as -3, -1, 1, 3, etc., while even integers can be divided evenly by 2, such as -4, -2, 0, 2, etc. Recognizing the properties of odd and even integers helps in determining the relationships between them, especially when finding consecutive odd integers, as they differ by 2.

Product of Integers

The product of two integers is the result of multiplying them together. In this problem, we are looking for two consecutive odd integers whose product equals 63. Understanding how to set up equations based on the product of integers is essential for solving the problem and finding the correct pairs.