Rewrite: \(x^2 - 5x = 2^3 = 8\).

["# Rewrite the Equation: Solve (x^2 - 5x = 8) with Confidence", "Solving quadratic equations is a fundamental skill in algebra, and one common challenge students face is rewriting equations in standard form before applying methods like factoring, completing the square, or using the quadratic formula. In this article, we’ll explore how to effectively rewrite the equation ( x^2 - 5x = 2^3 = 8 ) into a properly structured quadratic form—and solve it step-by-step.", "---", "## Why Rewriting Equations Matters", "Before solving, it’s essential to rewrite the equation in the standard quadratic form:\n[\nax^2 + bx + c = 0\n]\nThis form allows you to apply the most efficient solving techniques reliably. The given equation\n[\nx^2 - 5x = 8\n]\ncontains a constant on one side but isn’t yet fully rearranged. Transforming it properly ensures accuracy and clarity in your solution process.", "---", "## Step-by-Step Rewriting", "We begin with:\n[\nx^2 - 5x = 8\n]", "To convert this to standard form, we subtract 8 from both sides:", "[\nx^2 - 5x - 8 = 0\n]", "Now, the equation is in standard quadratic form:\n[\nx^2 - 5x - 8 = 0\n]\nwith coefficients:\n- (a = 1),\n- (b = -5),\n- (c = -8).", "---", "## Solving the Quadratic Equation", "With the equation ready, you can choose from multiple solution methods:", "### 1. Factoring (if possible)\nTry factoring (x^2 - 5x - 8 = 0). Unfortunately, this quadratic does not factor nicely with integers. So, we proceed to other methods.", "### 2. Quadratic Formula\nUse the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nSubstitute (a = 1), (b = -5), and (c = -8):", "[\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]", "Thus, the solutions are:\n[\nx = \frac{5 + \sqrt{57}}{2} \quad \ ext{and} \quad x = \frac{5 - \sqrt{57}}{2}\n]", "### 3. Completing the Square\nWe can also rewrite the original equation by completing the square. Starting with:\n[\nx^2 - 5x = 8\n]\nTake half of the coefficient of (x), square it, and add to both sides:\n[\nx^2 - 5x + \left(\frac{5}{2}\right)^2 = 8 + \left(\frac{5}{2}\right)^2\n]\n[\nx^2 - 5x + \frac{25}{4} = 8 + \frac{25}{4} = \frac{57}{4}\n]\nSo:\n[\n\left(x - \frac{5}{2}\right)^2 = \frac{57}{4}\n]\nTaking square roots:\n[\nx - \frac{5}{2} = \pm \frac{\sqrt{57}}{2}\n]\n[\nx = \frac{5 \pm \sqrt{57}}{2}\n]\nThis confirms our earlier results.", "---", "## Practical Tips for Rewriting Equations", "- Always move all terms to one side to form a zero on one side.\n- Use parentheses to keep expression clear.\n- Double-check arithmetic when calculating coefficients and discriminants.\n- When factoring is difficult, fall back on the quadratic formula or completing the square.", "---", "## Conclusion", "Rewriting (x^2 - 5x = 8) into standard form is a critical first step in solving quadratic equations. With the correct rearrangement, you can confidently apply computation methods like the quadratic formula or completing the square. Understanding this process not only helps solve this specific problem but builds a strong foundation for mastering algebra.", "Ready to practice? Try rewriting and solving another quadratic equation today, and watch your algebra skills grow.", "---\nKeywords: rewrite quadratic equation, solve (x^2 - 5x = 8), quadratic formula, completing the square, algebra solutions, step-by-step math, standard form algebra, quadratic problems."]