Let the integers be x and x+2. Then, x(x+2) = 143.

["Title: Solving x(x + 2) = 143: A Step-by-Step Guide to Integer Solutions", "Are you looking to solve the equation ( x(x + 2) = 143 ) in a clear and logical way? This article guides you through transforming boolean statements into real integers using algebra, revealing the integer solutions ( x = 11 ) and ( x = -13 ). Whether you're a student, teacher, or math enthusiast, understanding how to approach this type of problem unlocks deeper problem-solving skills.", "---", "### Understanding the Equation", "Let’s begin with the given equation:", "[\nx(x + 2) = 143\n]", "This statement uses integers ( x ) and ( x + 2 )—two consecutive odd (or even) integers if ( x ) is odd (or even). The equation expresses that the product of these two consecutive integers equals 143. Our goal is to find which integers satisfy this condition.", "---", "### Rearranging the Equation", "First, expand the left-hand side:", "[\nx(x + 2) = x^2 + 2x\n]", "Substitute into the equation:", "[\nx^2 + 2x = 143\n]", "Bring all terms to one side to form a standard quadratic equation:", "[\nx^2 + 2x - 143 = 0\n]", "---", "### Solving the Quadratic Equation", "Now we solve the quadratic equation ( x^2 + 2x - 143 = 0 ) using either factoring, completing the square, or the quadratic formula. Since factoring may not be straightforward, we use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 1 ), ( b = 2 ), ( c = -143 ). Plug in the values:", "[\nx = \frac{-2 \pm \sqrt{2^2 - 4(1)(-143)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 572}}{2} = \frac{-2 \pm \sqrt{576}}{2}\n]", "[\n\sqrt{576} = 24 \quad \Rightarrow \quad x = \frac{-2 \pm 24}{2}\n]", "This gives two solutions:", "[\nx = \frac{-2 + 24}{2} = \frac{22}{2} = 11\n]\n[\nx = \frac{-2 - 24}{2} = \frac{-26}{2} = -13\n]", "---", "### Interpreting the Integer Solutions", "So, the two integer solutions are:", "- ( x = 11 \Rightarrow x + 2 = 13 ), and indeed ( 11 \ imes 13 = 143 )\n- ( x = -13 \Rightarrow x + 2 = -11 ), and ( (-13) \ imes (-11) = 143 )", "These pairs ( (11, 13) ) and ( (-13, -11) ) are two sets of consecutive odd integers that multiply to 143.", "---", "### Why This Method Works", "By representing the integers as ( x ) and ( x+2 ), we modeled their relationship algebraically before transforming the word problem into a solvable equation. This approach demonstrates how real-world patterns (like consecutive integers) can be translated into mathematical expressions—ideal for strengthening algebraic reasoning.", "---", "### Real-World Applications & Extensions", "Understanding such equations helps with:", "- Cryptography and coding algorithms\n- Financial modeling with quadratic growth\n- Solving word problems in competitions and standardized tests\n- Developing critical thinking skills in STEM education", "---", "### Final Summary", "Solving ( x(x + 2) = 143 ) leads us to two integer solutions: ( x = 11 ) and ( x = -13 ). These values reflect two pairs of consecutive odd integers whose product is 143. This method—not only verifies the solution but deepens your ability to translate language into math and solve real-life problems with algebra.", "---", "Related Keywords: \nSolve quadratic equations, #Find integer solutions, #Algebra problems, #Consecutive integers equation, #Math tutorial, #Quadratic formula application, #Word problems explained", "---", "Keywords for SEO Optimization:\nLet integers be x and x+2, solve x(x+2) = 143, quadratic equation solution, integer solutions method, consecutive integers problem, algebra exercises with real-world context, step-by-step equation solving.", "---", "By mastering this approach, you empower yourself to tackle increasingly complex equations with clarity and confidence. Start with variables defined, solve systematically, and always verify your solutions—just as shown here with ( x(x + 2) = 143 )."]