Solve quadratic: \(x^2 - 3x - 8 = 0\)

["Solve Quadratic Equations: How to Solve (x^2 - 3x - 8 = 0)", "When it comes to solving quadratic equations, understanding methods like factoring, completing the square, or using the quadratic formula is essential. One common quadratic equation students encounter is:", "[\nx^2 - 3x - 8 = 0\n]", "In this article, we’ll walk through how to solve this equation step-by-step, along with explanations of key concepts and tips for quick problem-solving.", "---", "### Understanding Quadratic Equations", "A quadratic equation has the standard form:", "[\nax^2 + bx + c = 0\n]", "For the equation (x^2 - 3x - 8 = 0), the coefficients are:\n- (a = 1)\n- (b = -3)\n- (c = -8)", "Because solving by factoring isn’t immediately obvious, we’ll apply two powerful methods: factoring (when possible), the quadratic formula, and completing the square.", "---", "### Method 1: Using the Quadratic Formula", "The quark formula is a reliable way to solve any quadratic equation:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Plug in (a = 1), (b = -3), (c = -8):", "1. Compute the discriminant ((D)):", "[\nD = b^2 - 4ac = (-3)^2 - 4(1)(-8) = 9 + 32 = 41\n]", "2. Since (D > 0), there are two distinct real solutions.", "3. Plug values into the quadratic formula:", "[\nx = \frac{-(-3) \pm \sqrt{41}}{2(1)} = \frac{3 \pm \sqrt{41}}{2}\n]", "Solutions:", "[\nx = \frac{3 + \sqrt{41}}{2} \quad \ ext{and} \quad x = \frac{3 - \sqrt{41}}{2}\n]", "---", "### Method 2: Factoring (Not Straightforward Here)", "Unlike simpler quadratics like (x^2 - 5x + 6 = 0), this equation doesn't factor neatly into integers. That’s why alternative methods like completing the square or the quadratic formula work better here.", "---", "### Method 3: Completing the Square", "We'll rewrite the equation (x^2 - 3x - 8 = 0) by completing the square.", "1. Move constant term to the other side:", "[\nx^2 - 3x = 8\n]", "2. Take half of the coefficient of (x) (which is (-3)), so half is (-\frac{3}{2}), then square it:\n[\n\left(-\frac{3}{2}\right)^2 = \frac{9}{4}\n]", "3. Add (\frac{9}{4}) to both sides:", "[\nx^2 - 3x + \frac{9}{4} = 8 + \frac{9}{4}\n]", "4. Simplify the right-hand side:", "[\nx^2 - 3x + \frac{9}{4} = \frac{32}{4} + \frac{9}{4} = \frac{41}{4}\n]", "5. Write the left side as a perfect square:", "[\n\left(x - \frac{3}{2}\right)^2 = \frac{41}{4}\n]", "6. Take square roots of both sides:", "[\nx - \frac{3}{2} = \pm \sqrt{\frac{41}{4}} = \pm \frac{\sqrt{41}}{2}\n]", "7. Solve for (x):", "[\nx = \frac{3}{2} \pm \frac{\sqrt{41}}{2} = \frac{3 \pm \sqrt{41}}{2}\n]", "Same result as with the quadratic formula!", "---", "### Final Answers", "The solutions to the equation (x^2 - 3x - 8 = 0) are:", "[\n\boxed{ x = \frac{3 + \sqrt{41}}{2} } \quad \ ext{and} \quad \boxed{ x = \frac{3 - \sqrt{41}}{2} }\n]", "---", "### Tips for Quick Solving", "- Check for factorable quadratics first—look for two numbers that multiply to (c) and add to (b).\n- When factoring isn’t easy, use the quadratic formula—it works every time.\n- Completing the square is powerful and visually clarifies vertex form.\n- Use a calculator for square roots, but always simplify radicals where possible.", "---", "### Related Search Terms", "- How to solve (x^2 - 3x - 8 = 0?)\n- Solve quadratic equation using quadratic formula\n- Solve (x^2 - 3x - 8 = 0) step by step\n- Quadratic formula examples\n- Factor quadratic equations 쉽게 (Easy factoring)\n- Completing the square explained", "---", "Understanding how to solve quadratic equations equips you with a foundational skill in algebra, essential for higher math and real-world modeling. Whether you use the quadratic formula, factoring, or completing the square, mastering these techniques boosts confidence and problem-solving speed.", "Start practicing with different quadratics to make solving them second nature!"]