Roots: \(x = \frac{4 \pm \sqrt{64}}{4}\).

["### Solving (x = \frac{4 \pm \sqrt{64}}{4}): A Step-by-Step Guide", "Understanding how to solve quadratic-like equations involving square roots is essential in algebra. One common expression to simplify and solve is:", "[\nx = \frac{4 \pm \sqrt{64}}{4}\n]", "In this article, we’ll explore how to interpret and solve this expression step-by-step, explaining key algebraic concepts along the way. Whether you're a student learning algebra or a reader brushing up on radicals, this guide will clarify how to find the exact and approximate values of (x).", "---", "### Breaking Down the Equation", "The given equation is:", "[\nx = \frac{4 \pm \sqrt{64}}{4}\n]", "This involves a binomial expression with a square root under the radical—a typical form you’ll encounter when working with quadratic equations or radical simplifications.", "First, simplify the square root:", "[\n\sqrt{64} = 8\n]", "Substituting it in gives:", "[\nx = \frac{4 \pm 8}{4}\n]", "This now expresses two possible values for (x), because of the (\pm) symbol: one corresponding to the plus and one to the minus sign.", "---", "### Solving for Both Roots", "We evaluate both cases separately:", "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]", "---", "### Final Solutions", "Therefore, the two solutions to the equation (x = \frac{4 \pm \sqrt{64}}{4}) are:", "[\nx = 3 \quad \ ext{or} \quad x = -1\n]", "These roots can also be interpreted as the solutions to the quadratic equation ( (x - 3)(x + 1) = 0 ), confirming their validity.", "---", "### Why This Formula Matters", "The expression (\frac{4 \pm \sqrt{64}}{4}) is a simplified form often seen when solving quadratics through factoring or completing the square. The square root term (\sqrt{64} = 8) arises directly from simplifying radicals, and dividing by 4 comes from rationalizing expressions after extracting the root.", "Understanding such forms helps in:", "- Solving equations efficiently without full quadratic formulas\n- Recognizing patterns in quadratic expressions\n- Simplifying complex algebraic expressions", "---", "### Summary", "- Start with the original: (x = \frac{4 \pm \sqrt{64}}{4})\n- Simplify (\sqrt{64} = 8)\n- Rewrite as (\frac{4 \pm 8}{4})\n- Evaluate both signs to find (x = 3) and (x = -1)", "Whether you’re solving quadratic equations or working with rational expressions, mastering how to simplify and evaluate expressions with square roots is a foundational skill in algebra.", "---", "### Frequently Asked Questions (FAQs)", "Q: Why do we use (\pm) with square roots?\nA: The (\pm) sign reminds us there are two solutions—one using addition, one using subtraction—both valid in equations derived from factoring or symmetry.", "Q: Can (\sqrt{64}) be simplified?\nA: Yes, since (64 = 8^2), we simplify (\sqrt{64}) to 8 for easier computation.", "Q: How is this linked to quadratics?\nA: This form often appears when expressions factor into binomials like ((x - 3)(x + 1)). Solving directly avoids expanding, saving time.", "Q: What are real-world applications?\nA: Quadratic solutions model projectile motion, optimization problems, and any scenario involving curved paths or peak values—making these math skills practically useful.", "---", "### Keywords for SEO:\nRoots formula, solve quadratic equations, simplifying square roots, (\frac{4 \pm \sqrt{64}}{4}), algebraic steps, radical expressions, quadratic solutions, algebra tutorial, math problem-solving, solving equations, simplifying radicals", "---", "By mastering expressions like ( x = \frac{4 \pm \sqrt{64}}{4} ), you strengthen your algebraic toolkit—ready to tackle equations with confidence."]