\(x = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}\)

["# Solving the Quadratic Equation: (x = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4})", "Understanding how to solve quadratic equations is fundamental in algebra, and one common method is by simplifying expressions involving square roots and rational expressions. This article breaks down and solves the equation:", "[\nx = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}\n]", "## Step 1: Simplify the Square Root", "Begin by simplifying (\sqrt{64}). Since (8^2 = 64), we know:", "[\n\sqrt{64} = 8\n]", "Substituting this into the original equation gives:", "[\nx = \frac{4 \pm 8}{4}\n]", "## Step 2: Evaluate the Two Possible Forms", "The expression (\frac{4 \pm 8}{4}) represents two scenarios—one using addition and one using subtraction:", "[\nx = \frac{4 + 8}{4} \quad \ ext{and} \quad x = \frac{4 - 8}{4}\n]", "### Case 1: Plus Sign", "[\nx = \frac{4 + 8}{4} = \frac{12}{4} = 3\n]", "### Case 2: Minus Sign", "[\nx = \frac{4 - 8}{4} = \frac{-4}{4} = -1\n]", "## Step 3: Final Solutions", "The solutions to the equation are therefore:", "[\n\boxed{x = 3} \quad \ ext{or} \quad \boxed{x = -1}\n]", "## Why This Method Works", "Solving quadratic equations using the square root simplifies complex roots into rational or simplified forms, making it easier to interpret roots visually on a number line or graph. The expressions like (\frac{4 \pm \sqrt{64}}{4}) often appear in quadratic solutions derived from the quadratic formula when discriminants simplify neatly, as here with (\sqrt{64} = 8).", "---", "## Tips for Working with This Form", "- Always simplify the square root before substituting: (\sqrt{64} = 8), not leaving it as (\sqrt{64}) in final steps.\n- Track both signs ((+) and (-)) to capture all solutions.\n- Express solutions as exact fractions or simplified decimals depending on context.", "## Conclusion", "Solving (x = \frac{4 \pm \sqrt{64}}{4}) leads cleanly to two solutions: (x = 3) and (x = -1). This method highlights the power of simplifying radicals and separating cases with (\pm) to find all possible values of (x). Mastering this allows you to confidently handle similar quadratic expressions in algebra and calculus.", "---", "Keywords: quadratic equation solution, solve (x = \frac{4 \pm \sqrt{64}}{4}), simplify (\sqrt{64}), rational expressions, algebraic methods, solving linear equations with squares, algebra tutorial."]