The product of two consecutive even integers is 168. Find the smaller integer.
["Finding Two Consecutive Even Integers: The Product is 168 – What Smaller Integer Are We Looking For?", "When faced with a problem like “the product of two consecutive even integers is 168,” it’s a classic algebraic challenge that blends logic with basic number theory. Solving this not only helps sharpen your math skills but also reveals a straightforward way to identify solutions among even numbers. In this article, we’ll walk through how to find the smaller of the two consecutive even integers whose product equals 168, and explain the reasoning behind the process.", "---", "### Understanding Consecutive Even Integers", "Consecutive even integers are even numbers that follow each other without any gaps — for example, 2 and 4, 4 and 6, 6 and 8, etc. Since even integers can be expressed as ( 2n ) and ( 2n + 2 ) for some integer ( n ), we can use algebra to model their product clearly.", "---", "### Setting Up the Equation", "Let the smaller even integer be:", "$$\nx\n$$", "Then the next consecutive even integer is:", "$$\nx + 2\n$$", "Their product is given to be 168, so the equation becomes:", "$$\nx(x + 2) = 168\n$$", "---", "### Expanding and Rearranging", "Expanding the left-hand side:", "$$\nx^2 + 2x = 168\n$$", "Subtract 168 from both sides to form a standard quadratic equation:", "$$\nx^2 + 2x - 168 = 0\n$$", "---", "### Solving the Quadratic Equation", "We solve this quadratic using factoring, completion of the square, or the quadratic formula. In this case, we factor:", "We need two numbers whose product is (-168) and whose sum is (2). These numbers are 14 and -12.", "So,", "$$\nx^2 + 14x - 12x - 168 = 0\n$$", "Group terms:", "$$\n(x^2 + 14x) - (12x + 168) = 0\n$$", "Factor:", "$$\nx(x + 14) - 12(x + 14) = 0\n$$", "$$\n(x - 12)(x + 14) = 0\n$$", "Set each factor equal to zero:", "- ( x - 12 = 0 ) → ( x = 12 )\n- ( x + 14 = 0 ) → ( x = -14 )", "Both 12 and -14 are valid even integers, and their consecutive counterparts – 14 and 16, or -14 and -12 – both satisfy the original condition:", "- ( 12 \ imes 14 = 168 )\n- ( (-14) \ imes (-12) = 168 )", "---", "### Identifying the Smaller Integer", "From both solutions:", "- Smaller integer is ( 12 ) or ( -14 )", "Since the problem doesn’t specify positivity, both solutions are mathematically correct — but in most contexts, especially when focusing on smallest magnitude or standard integer sequences, the positive pair is preferred unless context demands the negative.", "For a clean, concise answer: The smaller of the two consecutive even integers is 12 (the smaller positive pair being typically emphasized unless otherwise noted).", "---", "### Final Thoughts", "Finding two consecutive even integers whose product is 168 is an elegant exercise in algebra and number patterns. By setting up a simple quadratic equation and solving for the integers, we confirm the smaller value is 12, proving that these integers are 12 and 14.", "Whether you're a student learning equations, a teacher illustrating problem-solving, or just a logic enthusiast, this classic problem highlights how to attack even (pun intended) challenges methodically.", "---", "Key Takeaways:\n- Represent consecutive even integers as ( x ) and ( x + 2 )\n- Form the equation ( x(x + 2) = 168 )\n- Solve the quadratic ( x^2 + 2x - 168 = 0 )\n- Integer solutions are ( 12 ) and ( -14 ), so the smaller is 12 (or ( -14 ))\n- Useful for building algebraic problem-solving skills", "If you found this helpful, share it to spread clear, step-by-step math learning!"]
