Use 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 \(x\). Since \(x\) is an odd integer, the next consecutive odd integer can be represented as \(x + 2\).
2. According to the problem, the product of these two consecutive odd integers is 63. So, we can write the equation: \(x \times (x + 2) = 63\).
3. Expand the left side of the equation to get a quadratic equation: \(x^2 + 2x = 63\).
4. Bring all terms to one side to set the equation equal to zero: \(x^2 + 2x - 63 = 0\).
5. Solve the quadratic equation \(x^2 + 2x - 63 = 0\) using factoring, completing the square, or the quadratic formula to find the values of \(x\), which represent the first odd integer.

Key Concepts

Consecutive Odd Integers

Consecutive odd integers are odd numbers that follow each other in order, each differing by 2. For example, if x is an odd integer, then x + 2 is the next consecutive odd integer. Understanding this helps in setting up expressions for problems involving sequences of odd numbers.

Algebraic Representation of Word Problems

Translating word problems into algebraic expressions involves defining variables to represent unknowns and using given relationships to form equations. Here, representing the two consecutive odd integers as x and x + 2 allows us to create an equation based on their product.

Solving Quadratic Equations

When the product of two expressions is given, setting up an equation often leads to a quadratic equation. Solving quadratic equations involves methods like factoring, completing the square, or using the quadratic formula to find the values of the variable that satisfy the equation.