Let the integers be \( x \) and \( x - 4 \), with \( x > 0 \), \( x(x - 4) = 120 \)

["Solving the Equation ( x(x - 4) = 120 ): A Step-by-Step Guide", "Struggling to solve quadratic equations in integer form? Want a clear, practical method for equations like ( x(x - 4) = 120 )? Look no further. In this SEO-optimized article, we’ll break down how to solve ( x(x - 4) = 120 ), justify why ( x ) and ( x - 4 ) are the right integers to consider, and explain how to find the positive solution efficiently.", "---", "### Understanding the Problem", "We are given:\n[ x(x - 4) = 120 ]\nwith the condition ( x > 0 ).\nOur goal is to find integer values satisfying this equation.", "First, rewrite the equation in standard quadratic form:\n[ x^2 - 4x = 120 ]\n[ x^2 - 4x - 120 = 0 ]", "This is a quadratic equation of the form ( ax^2 + bx + c = 0 ), where ( a = 1 ), ( b = -4 ), ( c = -120 ).", "---", "### Why Choose ( x ) and ( x - 4 )?", "Given the structure ( x ) and ( x - 4 ), it’s natural to define the two integers as consecutive-like values: ( x ) and the next smaller integer down by 4. This pattern is helpful because:", "- It reduces the problem to solving for ( x ) directly.\n- The product relationship ( x(x - 4) = 120 ) becomes easy to manipulate algebraically.\n- Using positive integers (( x > 0 )) ensures realistic, meaningful solutions in context (e.g., measurements, counts).", "---", "### Step 1: Rewrite the Equation", "Start from:\n[ x(x - 4) = 120 ]", "Expand:\n[ x^2 - 4x = 120 ]", "Bring all terms to one side:\n[ x^2 - 4x - 120 = 0 ]", "---", "### Step 2: Solve the Quadratic Equation", "Use the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plug in ( a = 1 ), ( b = -4 ), ( c = -120 ):\n[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-120)}}{2(1)}\n]\n[\nx = \frac{4 \pm \sqrt{16 + 480}}{2}\n]\n[\nx = \frac{4 \pm \sqrt{496}}{2}\n]", "Simplify ( \sqrt{496} ):\n[\n496 = 16 \ imes 31 \Rightarrow \sqrt{496} = 4\sqrt{31}\n]", "So,\n[\nx = \frac{4 \pm 4\sqrt{31}}{2} = 2 \pm 2\sqrt{31}\n]", "This gives two solutions:\n- ( x = 2 + 2\sqrt{31} )\n- ( x = 2 - 2\sqrt{31} )", "But only the first is positive (since ( \sqrt{31} > 5.5 ), so ( 2 - 2\sqrt{31} < 0 )).", "However, this form involves irrational numbers—defeating the purpose of integer solutions. This suggests we should avoid pixelating at the quadratic formula and instead leverage the integer constraint directly.", "---", "### Step 3: Factor the Quadratic Using Integer Insight", "Go back to:\n[ x^2 - 4x - 120 = 0 ]", "We want to factor this into two integers whose product is ( -120 ) and difference is ( -4 ) (from ( x \cdot (x - 4) = 120 ), i.e., one factor is 4 larger than the other in magnitude).", "List factor pairs of 120:\n- ( (1, 120), (2, 60), (3, 40), (4, 30), (5, 24), (6, 20), (8, 15), (10, 12) )", "Now, try pairs ( (a, b) ) such that ( a - b = 4 ) or ( b - a = 4 ), with one positive and one negative (since product is negative).", "Try:\n( x - (x - 4) = 4 ), so suppose ( x = a ), ( x - 4 = b ), with ( a \cdot b = 120 ).", "Try the pair ( (12, 10) ):\n( 12 \ imes 10 = 120 ), and ( 12 - 10 = 2 ) — too small.", "Try ( (15, 8) ):\n( 15 \cdot 8 = 120 ), difference = 7 — no.", "Try ( (20, 6) ):\n( 20 \cdot 6 = 120 ), difference = 14 — too big.", "Try reversing:\nLet ( x = 15 ), then ( x - 4 = 11 ): ( 15 \cdot 11 = 165 <br/>\ne 120 )\nTry ( x = 16 ): ( 16 \cdot 12 = 192 ) — too big.", "Wait — reconsider: since ( x(x - 4) = 120 ), and ( x > 0 ), try trial with integers near square root of 120 (~10.95).", "Try ( x = 16 ):\n( 16 \cdot (16 - 4) = 16 \cdot 12 = 192 ) — too big\nTry ( x = 15 ): ( 15 \cdot 11 = 165 ) — still big\nTry ( x = 14 ): ( 14 \cdot 10 = 140 )\nTry ( x = 12 ): ( 12 \cdot 8 = 96 ) — too small\nTry ( x = 13 ): ( 13 \cdot 9 = 117 )\nTry ( x = 14 ): already 140 — jump overshot.", "Wait: ( x = 16 ) too big, ( x = 15 ): 165 — problem.", "But earlier algebra gave irrational roots — so perhaps no integer solution?", "Wait—double-check:", "We want:\n[ x(x - 4) = 120 ]\nTry small positive integers greater than 4:", "- ( x = 6 ): ( 6 \cdot 2 = 12 )\n- ( x = 8 ): ( 8 \cdot 4 = 32 )\n- ( x = 10 ): ( 10 \cdot 6 = 60 )\n- ( x = 12 ): ( 12 \cdot 8 = 96 )\n- ( x = 14 ): ( 14 \cdot 10 = 140 )\n- ( x = 13 ): ( 13 \cdot 9 = 117 )\n- ( x = 16 ): ( 16 \cdot 12 = 192 )", "None yield 120.", "But earlier quadratic solution:\n[ x = \frac{4 \pm \sqrt{496}}{2} \approx \frac{4 \pm 22.27}{2} ]\n→ ( x \approx 13.13 ), not integer.", "So no integer solution.", "But the equation ( x(x - 4) = 120 ) has no integer solution?", "Wait — reconsider the problem statement.", "Ah! The equation does not have integer solutions, but we’re asked to “let the integers be ( x ) and ( x - 4 ), with ( x > 0 ), ( x(x - 4) = 120 )” — implying we solve for ( x ), assuming it’s available.", "But since no integer ( x ) satisfies it, perhaps the intention is to solve ( x(x - 4) = -120 )? No — problem says 120.", "Wait — mistake in factoring direction?", "Wait: ( x(x - 4) = 120 ), so ( x^2 - 4x - 120 = 0 ), discriminant ( 16 + 480 = 496 ), not a perfect square — no rational (hence no integer) solution.", "But perhaps the question wants us to set up and solve algebraically, even if no integer solution exists?", "But it says “with ( x(x - 4) = 120 )” and asks to “let the integers be ( x ) and ( x - 4 )” — so likely expects solving the quadratic and interpreting the result.", "Alternatively, reframe: maybe the problem is to find positive integer ( x ) satisfying ( x(x - 4) = 120 ) — and since none exist, state that.", "But that defeats the “solve” intent.", "Better interpretation: solve the equation, and present roots.", "---", "### Best Approach: Solve Quadratically — Present All Solutions", "From earlier:\n[ x = 2 \pm \sqrt{31} ]\nSince ( x > 0 ), we take:\n[ x = 2 + \sqrt{31} \approx 2 + 5.567 = 7.567 ]", "But not integer.", "So no positive integer solution exists.", "But if the problem expects factoring or algebraic manipulation with integer insight, then:", "Let’s instead suppose the equation is ( x(x - 4) = -120 ) — but it’s not."]