\( x = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4} \).
["# Solving the Equation ( x = \frac{3 \pm \sqrt{49}}{4} ): A Step-by-Step Explanation", "Solving quadratic expressions often presents a challenge, but with the right approach, breaking it down step-by-step makes the process simple and straightforward. One commonly encountered expression is:", "[\nx = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4}\n]", "In this article, we’ll explore how to simplify and solve this equation clearly and thoroughly, ideal for students, math enthusiasts, or anyone learning algebra.", "---", "## Step 1: Simplify the Square Root", "Start by recognizing that ( \sqrt{49} = 7 ), since 7 × 7 = 49. Substituting this into the equation gives:", "[\nx = \frac{3 \pm 7}{4}\n]", "Now we have two possible expressions depending on the ± sign:", "[\nx = \frac{3 + 7}{4} \quad \ ext{or} \quad x = \frac{3 - 7}{4}\n]", "---", "## Step 2: Solve Each Case Separately", "### Case 1: Adding 3 and 7", "[\nx = \frac{3 + 7}{4} = \frac{10}{4}\n]", "Simplify the fraction:", "[\nx = \frac{10 \div 2}{4 \div 2} = \frac{5}{2}\n]", "### Case 2: Subtracting 7 from 3", "[\nx = \frac{3 - 7}{4} = \frac{-4}{4}\n]", "Simplify the fraction:", "[\nx = -1\n]", "---", "## Step 3: Final Solutions", "From the two cases, the full set of solutions is:", "[\nx = \frac{5}{2} \quad \ ext{or} \quad x = -1\n]", "---", "## Why Understanding This Matters", "Solving equations of this form helps build a strong foundation in algebra, particularly in manipulating radical expressions and working with both positive and negative results. These skills are essential for advanced math topics like quadratic equations, graphing functions, and systems of equations.", "---", "## Key Takeaways", "- Simplifying radicals first makes solving easier.\n- The ± symbol leads to two distinct solutions.\n- Breaking each expression into separate cases ensures accuracy.\n- Final simplified answers are ( x = \frac{5}{2} ) and ( x = -1 ).", "---", "### Practice Problems", "Try solving similar expressions:", "[\nx = \frac{1 \pm \sqrt{16}}{5}, \quad x = \frac{4 \pm \sqrt{25}}{6}, \quad x = \frac{-2 \pm \sqrt{9}}{3}\n]", "With practice, simplifying these will feel natural and quick!", "---", "### Conclusion", "Mastering equations like ( x = \frac{3 \pm \sqrt{49}}{4} ) involves recognizing perfect squares, simplifying fractions, and carefully handling both positive and negative scenarios. By following a structured step-by-step method, you can solve similar algebraic expressions with confidence and precision. Keep practicing—algebra is a skill that grows stronger with every problem solved!", "---", "Keywords: algebra, solving quadratic equations, rational expressions, simplifying radicals, fraction simplification, ± rule in equations, step-by-step solving, quadratic equation practice, mathematical problem solving.", "---", "Meta Title: How to Solve ( x = \frac{3 \pm \sqrt{49}}{4} ): Step-by-Step Guide\nMeta Description: Learn to simplify and solve ( x = \frac{3 \pm \sqrt{49}}{4} ) using clear mathematical steps, including radical simplification and case analysis. Ideal for students and learners."]
