A quadratic equation \( x^2 - 4x - 5 = 0 \) has solutions:

["# Quadratic Equation ( x^2 - 4x - 5 = 0 ): How to Solve It and Find Its Solutions", "Solving quadratic equations is a foundational skill in algebra, essential for both academic success and real-world problem solving. One classic example is the equation:", "[ x^2 - 4x - 5 = 0 ]", "Understanding how to solve this equation using standard methods helps build confidence for more complex problems. In this article, we explore how to find the solutions to this quadratic equation step-by-step, explain the key concepts, and discuss the importance of quadratic equations in mathematics and science.", "---", "## What Is a Quadratic Equation?", "A quadratic equation is any equation of the form:", "[ ax^2 + bx + c = 0 ]", "where ( a ), ( b ), and ( c ) are constants, and ( a <br/>\neq 0 ). The general solution involves using the quadratic formula:", "[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]", "This formula provides two solutions: one using the plus sign and one using the minus sign, corresponding to the two roots of the equation.", "---", "## Step-by-Step Solution of ( x^2 - 4x - 5 = 0 )", "### Step 1: Identify coefficients\nFrom the equation ( x^2 - 4x - 5 = 0 ), we identify:\n- ( a = 1 )\n- ( b = -4 )\n- ( c = -5 )", "### Step 2: Calculate the discriminant\nThe discriminant ( D = b^2 - 4ac ) determines the nature of the solutions.\n[\nD = (-4)^2 - 4(1)(-5) = 16 + 20 = 36\n]\nSince ( D > 0 ) and a perfect square, we know there are two distinct real solutions.", "### Step 3: Apply the quadratic formula\n[\nx = \frac{-(-4) \pm \sqrt{36}}{2 \ imes 1} = \frac{4 \pm 6}{2}\n]", "### Step 4: Compute both solutions\n[\nx_1 = \frac{4 + 6}{2} = \frac{10}{2} = 5\n]\n[\nx_2 = \frac{4 - 6}{2} = \frac{-2}{2} = -1\n]", "---", "## Final Answer", "The solutions to the quadratic equation ( x^2 - 4x - 5 = 0 ) are:\n[\n\boxed{x = -1 \quad} \ ext{and} \quad \boxed{x = 5}\n]", "---", "## Why Are Quadratic Equations Important?", "Quadratic equations appear frequently in physics, engineering, economics, and computer science. They model projectile motion, optimize areas and profits, describe electrical circuits, and define parabolic curves in design and graphics. Mastering their solution is crucial for students and professionals alike.", "---", "## How to Check Your Solutions", "You can verify the solutions by substituting ( x = 5 ) and ( x = -1 ) back into the original equation:", "- For ( x = 5 ):\n ( (5)^2 - 4(5) - 5 = 25 - 20 - 5 = 0 ) ✅\n- For ( x = -1 ):\n ( (-1)^2 - 4(-1) - 5 = 1 + 4 - 5 = 0 ) ✅", "Both values satisfy the equation.", "---", "## Conclusion", "Finding the solutions to ( x^2 - 4x - 5 = 0 ) gives ( x = -1 ) and ( x = 5 ), derived using the quadratic formula and confirmed through algebraic verification. This example illustrates the power of a standard formula and reinforces key algebraic techniques essential for advancing in mathematics and related disciplines.", "If you're learning quadratics and need practice, try solving similar equations with different coefficients—each offers valuable reinforcement of the method.", "---", "Keywords: quadratic equation, solve ( x^2 - 4x - 5 = 0 ), quadratic formula, real solutions, algebra, step-by-step solution, discriminant, math tutorial, math problems, interactive math, quadratic roots, solve quadratic equations", "Meta Description: Learn how to solve the quadratic equation ( x^2 - 4x - 5 = 0 ) step-by-step using the quadratic formula, including solution verification and real-world relevance. Perfect for students and math enthusiasts!"]