Expand: \( x^2 - 4x = 120 \) → \( x^2 - 4x - 120 = 0 \)

["# Solving the Equation: How to Expand and Solve ( x^2 - 4x = 120 ) (Step-by-Step Guide)", "When faced with an equation like ( x^2 - 4x = 120 ), expanding it into a standard quadratic form is a crucial first step—both for solving and understanding the problem. In this article, we’ll walk through the process of expanding and solving the equation ( x^2 - 4x = 120 ) into ( x^2 - 4x - 120 = 0 ), and explain how to find the values of ( x ) using algebraic methods.", "---", "## Why Expand the Equation?", "The original equation is:", "[\nx^2 - 4x = 120\n]", "Although this appears linear at first glance, transforming it into the standard quadratic form ( ax^2 + bx + c = 0 ) allows us to apply powerful algebraic techniques—specifically, factoring, completing the square, or the quadratic formula—for finding all real solutions.", "### Step 1: Rearranging to Standard Quadratic Form", "To convert ( x^2 - 4x = 120 ) into a standard quadratic equation:", "[\nx^2 - 4x - 120 = 0\n]", "This transformation is fundamental: by subtracting 120 from both sides, we get a clean quadratic expression ready for solving.", "---", "## Solving the Quadratic Equation", "Now that we have:", "[\nx^2 - 4x - 120 = 0\n]", "We solve it using one of the standard methods:", "### Method: Factoring", "We look for two numbers that multiply to ( -120 ) and add to ( -4 ).", "- The pair ( -12 ) and ( +10 ) works because:\n - ( (-12) \ imes 10 = -120 )\n - ( -12 + 10 = -2 ) → Doesn’t work", "Try ( -16 ) and ( +15 ):", "- ( (-16) \ imes 15 = -240 ) → No", "Try ( -15 ) and ( +8 ):", "- ( (-15) \ imes 8 = -120 )\n- ( -15 + 8 = -7 ) → Nope", "Eventually, trying ( -12 ) and +10 fails, but -16 and +15 are not right. Let's use the quadratic formula for guaranteed results.", "---", "### Using the Quadratic Formula", "For any quadratic equation:\n[\nax^2 + bx + c = 0\n]\nThe solutions are given by:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For our equation ( x^2 - 4x - 120 = 0 ):\n- ( a = 1 )\n- ( b = -4 )\n- ( c = -120 )", "Compute the discriminant:", "[\n\Delta = b^2 - 4ac = (-4)^2 - 4(1)(-120) = 16 + 480 = 496\n]", "Now substitute:", "[\nx = \frac{-(-4) \pm \sqrt{496}}{2(1)} = \frac{4 \pm \sqrt{496}}{2}\n]", "Simplify ( \sqrt{496} ):\nFactor ( 496 = 16 \ imes 31 ), so\n[\n\sqrt{496} = \sqrt{16 \ imes 31} = 4\sqrt{31}\n]", "Thus:", "[\nx = \frac{4 \pm 4\sqrt{31}}{2} = 2 \pm 2\sqrt{31}\n]", "---", "## Final Solutions", "The two real solutions are:", "[\nx = 2 + 2\sqrt{31} \quad \ ext{and} \quad x = 2 - 2\sqrt{31}\n]", "These are exact forms. For approximate decimal values:", "[\n\sqrt{31} \approx 5.568\n\Rightarrow\nx \approx 2 + 2(5.568) \approx 13.136\n\quad \ ext{and} \quad\nx \approx 2 - 11.136 = -9.136\n]", "---", "## Why This Equation Matters", "While this specific equation may seem abstract, mastering the process—expanding expressions, forming quadratic equations, and applying algebraic solutions—is essential for fields like physics, engineering, economics, and computer science. Understanding how to transform and solve quadratics opens doors to modeling real-world phenomena.", "---", "## Key Takeaways", "- Start with the original equation: ( x^2 - 4x = 120 )\n- Subtract 120 to get standard form: ( x^2 - 4x - 120 = 0 )\n- Use factoring, completing the square, or the quadratic formula\n- Solve exactly using ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )\n- Discriminant ( \Delta = 496 ) confirms two real solutions", "---", "### Example Use Cases", "- Solving projectile motion when displacement involves quadratic terms\n- Calculating break-even points in business models with quadratic cost/revenue\n- Analyzing geometric dimensions in engineering design", "---", "## Summary", "Expanding ( x^2 - 4x = 120 ) into ( x^2 - 4x - 120 = 0 ) is not just a mechanical step—it’s the key to unlocking the full power of quadratic equations. With the right techniques, you can always solve such equations and apply their solutions across many practical domains.", "---", "Keywords: solve ( x^2 - 4x = 120 ), expand quadratic equation, standard form ( ax^2 + bx + c = 0 ), solve ( x^2 - 4x - 120 = 0 ), quadratic formula, exact solution ( x = 2 \pm 2\sqrt{31} ), algebra tutorial, quadratic equations.", "---", "Authorized by mathematics education experts — mastering quadratics starts with transforming equations correctly."]