Expand: \( x^2 - 3x - 4 = 0 \).

["# Solving the Quadratic Equation ( x^2 - 3x - 4 = 0 ): Step-by-Step Guide", "When faced with the quadratic equation:", "[\nx^2 - 3x - 4 = 0\n]", "solving for ( x ) is a fundamental algebraic skill with broad applications in mathematics, physics, engineering, and economics. This article provides a clear, detailed walkthrough of how to expand, factor, and solve the equation, along with verification and real-world relevance.", "---", "## Step 1: Understand the Standard Form", "A quadratic equation in two variables is generally expressed as:", "[\nax^2 + bx + c = 0\n]", "Comparing this to ( x^2 - 3x - 4 = 0 ), we identify:\n- ( a = 1 )\n- ( b = -3 )\n- ( c = -4 )", "---", "## Step 2: Expand and Factor the Quadratic Equation", "Although this equation is already in standard form, it can be viewed as a product of binomials — factoring is one of the most efficient solving methods. We aim to express it as:", "[\n(x + m)(x + n) = 0\n]", "To factor ( x^2 - 3x - 4 ), we look for two numbers that:\n- Multiply to ( c = -4 ) (constant term)\n- Add to ( b = -3 ) (linear coefficient)", "After testing integer pairs, we find:\n- ( 1 \ imes (-4) = -4 )\n- ( 1 + (-4) = -3 ) ✓", "So, the factored form is:", "[\n(x + 1)(x - 4) = 0\n]", "---", "## Step 3: Apply the Zero Product Property", "Since the product of two factors is zero, at least one factor must be zero:", "[\nx + 1 = 0 \quad \ ext{or} \quad x - 4 = 0\n]", "Solving each:", "- ( x = -1 )\n- ( x = 4 )", "---", "## Step 4: Verify the Solutions", "Substitute ( x = -1 ) and ( x = 4 ) back into the original equation:", "For ( x = -1 ):", "[\n(-1)^2 - 3(-1) - 4 = 1 + 3 - 4 = 0 \quad \ ext{✓}\n]", "For ( x = 4 ):", "[\n(4)^2 - 3(4) - 4 = 16 - 12 - 4 = 0 \quad \ ext{✓}\n]", "Both solutions are valid.", "---", "## Why Solving ( x^2 - 3x - 4 = 0 ) Matters", "Understanding how to solve quadratic equations equips learners with essential techniques for modeling real-world scenarios such as:", "- Projectile motion in physics\n- Revenue and cost optimization in business\n- Area and perimeter problems in geometry\n- Any situation modeled by a second-degree relationship", "---", "## Additional Methods: Using the Quadratic Formula", "Though factoring is straightforward here, the quadratic formula works universally:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plugging in ( a = 1 ), ( b = -3 ), ( c = -4 ):", "[\nx = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)} = \frac{3 \pm \sqrt{9 + 16}}{2} = \frac{3 \pm \sqrt{25}}{2} = \frac{3 \pm 5}{2}\n]", "So:", "- ( x = \frac{3 + 5}{2} = 4 )\n- ( x = \frac{3 - 5}{2} = -1 )", "Confirming the same solutions efficiently.", "---", "## Conclusion", "The quadratic equation ( x^2 - 3x - 4 = 0 ) exemplifies core algebraic techniques involving expansion, factoring, and root-finding. Mastering these methods strengthens mathematical reasoning and opens the door to solving complex equations in science and technology. Regular practice with such equations ensures faster recognition and boosts confidence in handling higher-level math.", "---", "## Keywords for SEO Optimization\n- Solve ( x^2 - 3x - 4 = 0 )\n- Quadratic equation solutions\n- Factoring quadratic equations\n- Algebraic methods for quadratic\n- Solve ( ax^2 + bx + c = 0 )\n- Important quadratic equation examples", "---", "Improve your quadratic equation proficiency today — start solving now!"]