So, \(x^2 - 5x - 8 = 0\).

["# Solving the Quadratic Equation: (x^2 - 5x - 8 = 0)", "Solving quadratic equations is a fundamental skill in algebra, essential for students, educators, and math enthusiasts. One of the most commonly encountered equations is (x^2 - 5x - 8 = 0). Whether you’re preparing for an exam or simply seeking to understand how to solve this type of problem, this guide breaks down the steps clearly and thoroughly.", "## What is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation of the form:", "[\nax^2 + bx + c = 0\n]", "where (a), (b), and (c) are constants, and (a <br/>\neq 0). The equation (x^2 - 5x - 8 = 0) fits this definition with (a = 1), (b = -5), and (c = -8).", "## Why Solve (x^2 - 5x - 8 = 0)?", "Understanding this equation helps develop problem-solving skills in algebra. It's a standard form that allows us to use factoring, completing the square, or the quadratic formula. Solving such equations is not only academically valuable but also practical in physics, engineering, and economics where quadratic models frequently appear.", "## Step-by-Step Solution", "There are three standard methods to solve quadratic equations—factoring, completing the square, and the quadratic formula. Let’s explore each method in detail using the equation:", "[\nx^2 - 5x - 8 = 0\n]", "### 1. Factoring Approach", "Factoring involves rewriting the quadratic into the product of two binomials. We look for two numbers that multiply to (c = -8) and add to (b = -5).", "- Factors of -8: (-8, 1), (8, -1), (-4, 2), (4, -2)\n- Among these, (-8) and (1) add to (-7) → not correct\n- (1) and (-8) → adds to (-7) → no match\n- Try (4) and (-2): sum = 2 → no\n- Try (8) and (-1): sum = 7 → no \nUnfortunately, no integer pair multiplies to -8 and adds to -5, meaning the expression does not factor nicely with integers.", "### 2. Quadratic Formula", "When factoring is difficult, the quadratic formula guarantees a solution:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For our equation (x^2 - 5x - 8 = 0):", "- (a = 1), (b = -5), (c = -8)", "Plug into the formula:", "[\nx = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-8)}}{2(1)} = \frac{5 \pm \sqrt{25 + 32}}{2} = \frac{5 \pm \sqrt{57}}{2}\n]", "So the solutions are:", "[\nx = \frac{5 + \sqrt{57}}{2} \quad \ ext{and} \quad x = \frac{5 - \sqrt{57}}{2}\n]", "### 3. Completing the Square (Alternative Method)", "This method transforms the equation into a perfect square trinomial.", "Starting with:", "[\nx^2 - 5x - 8 = 0\n]", "Move the constant:", "[\nx^2 - 5x = 8\n]", "Take half of the coefficient of (x), which is (-5/2), and square it: ((-5/2)^2 = 25/4). Add this to both sides:", "[\nx^2 - 5x + \frac{25}{4} = 8 + \frac{25}{4}\n]", "Left side is now a perfect square:", "[\n\left(x - \frac{5}{2}\right)^2 = \frac{32}{4} + \frac{25}{4} = \frac{57}{4}\n]", "Take square roots:", "[\nx - \frac{5}{2} = \pm \frac{\sqrt{57}}{2}\n]", "Solve for (x):", "[\nx = \frac{5}{2} \pm \frac{\sqrt{57}}{2} = \frac{5 \pm \sqrt{57}}{2}\n]", "Same results as with the quadratic formula.", "## Final Results & Interpretation", "The two real solutions to (x^2 - 5x - 8 = 0) are:", "[\nx = \frac{5 + \sqrt{57}}{2} \quad \ ext{and} \quad x = \frac{5 - \sqrt{57}}{2}\n]", "Approximate decimal values are:", "[\nx \approx \frac{5 + 7.5498}{2} \approx 6.2749 \quad \ ext{and} \quad x \approx \frac{5 - 7.5498}{2} \approx -1.2749\n]", "## Why Use the Quadratic Formula?", "While factoring works neatly in many cases, not all quadratics factor easily. The quadratic formula provides a universal method applicable to all quadratic equations, ensuring accurate solutions through algebraic manipulation.", "## Applications of (x^2 - 5x - 8 = 0)", "This equation models real-world scenarios such as:", "- Projectile motion: predicting where a thrown object will land\n- Business: maximizing profit or minimizing cost functions\n- Engineering: analyzing structural loads and forces", "Solving it helps students and professionals analyze and predict outcomes in dynamic systems.", "## Summary", "- (x^2 - 5x - 8 = 0) is a standard quadratic equation\n- It cannot be factored neatly with integers, but the quadratic formula applies\n- Two real, irrational solutions exist: (\frac{5 \pm \sqrt{57}}{2})\n- The equation models many practical applications across disciplines", "### Tips for Mastering Quadratic Equations\n- Practice identifying coefficients quickly\n- Memorize the quadratic formula and when to use each solving method\n- Use graphing tools to visualize roots and verify solutions\n- Apply the equation in word problems to reinforce understanding", "---", "Whether you’re learning algebra for the first time or brushing up on fundamentals, solving (x^2 - 5x - 8 = 0) sharpens critical reasoning and problem-solving skills essential in both academic and real-life contexts. Start practicing today—your next victory in algebra begins now!"]