The product of two consecutive even integers is 168. What is the larger integer?
["The Product of Two Consecutive Even Integers is 168: Find the Larger Integer", "Are you curious about how math connects with real-life problem-solving? One common challenge students face is finding two consecutive even integers whose product equals a given number—like 168. In this article, we’ll uncover the solution step-by-step and reveal the larger of the two even integers.", "### What Are Consecutive Even Integers?", "Consecutive even integers are two even numbers that follow one after the other in the sequence of even numbers. For example, 4 and 6, 10 and 12, or — in this case — two unknown even integers ( x ) and ( x + 2 ).", "### Setting Up the Equation", "Let ( x ) be the smaller even integer. Then the next consecutive even integer is ( x + 2 ). Their product is given:", "[\nx(x + 2) = 168\n]", "This expands to:", "[\nx^2 + 2x = 168\n]", "Rearranging into standard quadratic form:", "[\nx^2 + 2x - 168 = 0\n]", "### Solving the Quadratic Equation", "We solve the quadratic equation using factoring, completing the square, or the quadratic formula. Since factoring is straightforward here, let’s find two numbers that multiply to (-168) and add to (2).", "Factors of 168:\n( 12 \ imes 14 = 168 ), and since (14 - 12 = 2), we use:", "[\n(x + 14)(x - 12) = 0\n]", "Setting each factor equal to zero:", "[\nx + 14 = 0 \quad \ ext{or} \quad x - 12 = 0\n]", "[\nx = -14 \quad \ ext{or} \quad x = 12\n]", "We have two possible solutions:\n- ( x = 12 ), so the integers are 12 and 14\n- ( x = -14 ), so the integers are –14 and –12", "Both pairs are valid consecutive even integers, but the question asks for the larger integer.", "### Identifying the Larger Integer", "- For ( 12 ) and ( 14 ), the larger integer is 14.\n- For ( -14 ) and ( -12 ), the larger is –12, which is less than 14.", "Thus, the larger integer is:", "[\n\boxed{14}\n]", "### Final Thoughts", "Finding consecutive even integers through algebra reinforces key math skills—factoring quadratics, reasoning about integer sequences, and solving real-world problems. Whether working on homework or tackling mathematic puzzles, understanding how to break down such problems made easy helps build confidence in math.", "Next time you see a product of two consecutive evens equal 168, remember: the larger integer is 14!", "---", "Keywords: consecutive even integers, product of two consecutive even integers is 168, solve quadratic equations, even integers, algebra problem, math tutorial, math practice, integer sequence, quadratic formula"]
