\(x = \frac{3 \pm \sqrt{9 + 16}}{4}\).

["# Solving the Equation: (x = \frac{3 \pm \sqrt{9 + 16}}{4})", "Understand the step-by-step solution to the algebraic expression (x = \frac{3 \pm \sqrt{9 + 16}}{4})—a common form in algebra that combines simplification, root evaluation, and divisor application.", "## Understanding the Equation", "The given expression is:\n[\nx = \frac{3 \pm \sqrt{9 + 16}}{4}\n]\nThis equation uses the ± (plus-minus) notation, indicating that there are two possible solutions—one with addition and one with subtraction—resulting from solving a quadratic or binomial simplification context.", "## Step 1: Simplify the Expression Under the Square Root", "First, simplify the expression under the square root:\n[\n9 + 16 = 25\n]\nSo the equation becomes:\n[\nx = \frac{3 \pm \sqrt{25}}{4}\n]", "## Step 2: Evaluate the Square Root", "Since (\sqrt{25} = 5), substitute:\n[\nx = \frac{3 \pm 5}{4}\n]", "## Step 3: Solve for Both Roots", "Now compute the two distinct solutions based on the ± operator:", "- First solution (plus case):\n[\nx = \frac{3 + 5}{4} = \frac{8}{4} = 2\n]", "- Second solution (minus case):\n[\nx = \frac{3 - 5}{4} = \frac{-2}{4} = -\frac{1}{2}\n]", "## Step 4: Final Answer", "Thus, the two solutions to the equation are:\n[\nx = 2 \quad \ ext{and} \quad x = -\frac{1}{2}\n]", "### Alternative Representation", "The two roots can also be written clearly as:\n[\nx_1 = 2, \quad x_2 = -\frac{1}{2}\n]", "## Why This Equation Matters", "This form frequently appears in algebra when solving quadratic equations derived from standard forms like (ax^2 + bx + c = 0), particularly when factoring or completing the square leads to a square root expression. It’s also essential for understanding function domains, solving equations in physics, and graphing scenarios involving symmetry about vertical lines.", "## Conclusion", "The equation (x = \frac{3 \pm \sqrt{9 + 16}}{4}) simplifies neatly to two real solutions:\n[\n\boxed{x = 2 \quad \ ext{and} \quad x = -\frac{1}{2}}\n]\nMastering such expressions strengthens foundational algebra skills crucial for advanced mathematics, science, and engineering disciplines.", "---", "Keywords: algebra solution, (\frac{3 \pm \sqrt{9 + 16}}{4}), quadratic roots, simplification, solving equations, ± in algebra, computer algebra system, math tutorial, educational resource", "Meta Description: Learn how to solve (x = \frac{3 \pm \sqrt{9 + 16}}{4}). Step-by-step breakdown of simplifying radicals, evaluating solutions, and verifying results in algebra."]