The sum of the squares of two consecutive even integers is 340. What are the integers?

["The Sum of the Squares of Two Consecutive Even Integers Equals 340\nDiscover the Consecutive Even Integers in Just Minutes", "If you're wondering which two consecutive even integers have a sum of squares equal to 340, you're in the right place! This classic problem combines algebra with number thinking, making it a perfect example to solve using both logic and mathematics. In this article, we’ll walk through how to find these integers step by step and explain why this approach works—ideal for students, math enthusiasts, and anyone learning problem-solving techniques.", "---", "### Understanding the Problem", "We are told that the sum of the squares of two consecutive even integers is 340. Let’s break this down:", "- Consecutive even integers mean two even numbers that follow each other without gaps. For example, 4 and 6, or 10 and 12.\n- Let the smaller even integer be ( x ).\n- Then, the next consecutive even integer is ( x + 2 ).\n- The sum of their squares is ( x^2 + (x + 2)^2 = 340 ).", "Our goal is to solve this equation and find the two integers.", "---", "### Setting Up the Equation", "Start by writing the equation based on the definition:", "[\nx^2 + (x + 2)^2 = 340\n]", "Now expand ( (x + 2)^2 ):", "[\nx^2 + (x^2 + 4x + 4) = 340\n]", "Combine like terms:", "[\n2x^2 + 4x + 4 = 340\n]", "Subtract 340 from both sides:", "[\n2x^2 + 4x + 4 - 340 = 0\n]", "[\n2x^2 + 4x - 336 = 0\n]", "---", "### Simplify the Equation", "Divide every term by 2 to simplify:", "[\nx^2 + 2x - 168 = 0\n]", "This is a quadratic equation in standard form. We can solve it using the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 1 ), ( b = 2 ), and ( c = -168 ). Plug in the values:", "[\nx = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-168)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 672}}{2} = \frac{-2 \pm \sqrt{676}}{2}\n]", "[\n\sqrt{676} = 26\n]", "So,", "[\nx = \frac{-2 \pm 26}{2}\n]", "This gives two possible solutions:", "1. ( x = \frac{-2 + 26}{2} = \frac{24}{2} = 12 )\n2. ( x = \frac{-2 - 26}{2} = \frac{-28}{2} = -14 )", "---", "### Find the Two Consecutive Even Integers", "- If ( x = 12 ), the next consecutive even integer is ( 12 + 2 = 14 )\n- If ( x = -14 ), then ( x + 2 = -12 )", "Let’s verify both pairs:", "Pair 1: 12 and 14\n[\n12^2 + 14^2 = 144 + 196 = 340 \quad \ ext{✓ Correct!}\n]", "Pair 2: -14 and -12\n[\n(-14)^2 + (-12)^2 = 196 + 196 = 392 \quad \ ext{✗ Incorrect!}\n]", "Wait—above, we incorrectly assumed that both pairs work. But hold on: they do not. Only one pair satisfies the original equation.", "Why? Because the equation ( x^2 + (x + 2)^2 = 340 ) reflects increasing even numbers from the lower value. While ( (-14)^2 + (-12)^2 = 392 ) exceeds 340, it corresponds to squaring larger magnitudes in negative numbers but gives a larger sum.", "But note: 392 > 340, so ( x = -14 ) leads to too large a sum.", "However, wait—let’s test small even numbers manually nearby to confirm:", "Try ( x = 10 ):\n( 10^2 + 12^2 = 100 + 144 = 244 )", "Too low.", "Try ( x = 12 ):\n( 12^2 + 14^2 = 144 + 196 = 340 ) ✓ Correct!", "Try ( x = 8 ):\n( 8^2 + 10^2 = 64 + 100 = 164 ) — still too low.", "So indeed, the only solution where the sum of squares equals 340 is ( 12 ) and ( 14 ).", "---", "### Why This Works: A Quick Explanation", "The sum of the squares of two consecutive even integers depends on their positions. Squaring an even number increases rapidly:\nEven numbers around 12 yield sum = 340, while those farther out (like 8 or 16) give too small or too large results. This reinforces why solving algebraically is reliable.", "---", "### Final Answer", "The two consecutive even integers whose squares sum to 340 are:", "[\n\boxed{12\ \ ext{and}\ 14}\n]", "---", "### Why You Should Learn This Problem", "Understanding how to translate word problems into quadratic equations empowers you to solve a wide range of mathematical and real-world challenges. Whether preparing for exams, teaching math, or simply sharpening logical reasoning, mastering such problems strengthens your analytical toolkit.", "---", "Keywords: sum of the squares of two consecutive even integers, quadratic equation, consecutive even integers, solve algebraically, math problem-solving, 340 sum of squares, even integers equation, algebraic solution, step-by-step math", "Meta Description: Find the two consecutive even integers whose squares add to 340. We solve the equation step-by-step and verify correctness—perfect for students and math learners.", "---", "Need more math practice? Explore related problems on consecutive numbers, quadratic equations, and integer solutions!"]