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 integers whose product is 110.

Verified step by step guidance

1. Let the first integer be represented by \(x\). Since the problem asks for two consecutive integers, the next consecutive integer can be represented as \(x+1\).
2. Write an equation for the product of these two consecutive integers. The product is given as 110, so the equation is \(x \times (x+1) = 110\).
3. Expand the left side of the equation using the distributive property: \(x^2 + x = 110\).
4. Rewrite the equation in standard quadratic form by subtracting 110 from both sides: \(x^2 + x - 110 = 0\).
5. Solve the quadratic equation \(x^2 + x - 110 = 0\) using factoring, completing the square, or the quadratic formula to find the values of \(x\) that satisfy the equation.

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in order, differing by 1. For example, if x is an integer, then x and x+1 are consecutive integers. Understanding this helps set up equations involving consecutive numbers.

Forming Algebraic Equations from Word Problems

Translating a word problem into an algebraic equation involves identifying variables and expressing relationships mathematically. Here, the product of two consecutive integers is given, so we write an equation like x(x+1) = 110 to solve for x.

Solving Quadratic Equations

When the product of two consecutive integers is set equal to a number, it forms a quadratic equation. Solving this involves expanding, rearranging, and using methods like factoring or the quadratic formula to find integer solutions.