4y^2 - 12y - 1 \leq 0

["# Solving the Quadratic Inequality: Understanding 4y² - 12y - 1 ≤ 0", "When tackling quadratic inequalities, understanding how to solve equations like 4y² - 12y - 1 ≤ 0 is essential for mastering algebra and applying it to real-world problems. Whether you're a student learning algebra or someone brushing up on key math concepts, this step-by-step guide will break down how to analyze and solve the inequality 4y² - 12y - 1 ≤ 0 effectively. Let’s dive in.", "---", "## What Is the Inequality?", "We begin with the quadratic inequality:\n4y² - 12y - 1 ≤ 0", "This means we’re looking for all real values of y where the quadratic expression 4y² - 12y - 1 is less than or equal to zero — in other words, where the graph of the parabola lies below or touches the x-axis.", "---", "## Step 1: Find the Roots of the Corresponding Equation", "First, solve the related quadratic equation:\n4y² - 12y - 1 = 0", "We apply the quadratic formula:\n[\ny = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nwhere ( a = 4 ), ( b = -12 ), and ( c = -1 ).", "Calculate the discriminant:\n[\nb^2 - 4ac = (-12)^2 - 4(4)(-1) = 144 + 16 = 160\n]\nSince the discriminant is positive, there are two distinct real roots.", "Now compute the roots:\n[\ny = \frac{12 \pm \sqrt{160}}{8} = \frac{12 \pm 4\sqrt{10}}{8} = \frac{3 \pm \sqrt{10}}{2}\n]", "So, the roots are:\n[\ny_1 = \frac{3 - \sqrt{10}}{2}, \quad y_2 = \frac{3 + \sqrt{10}}{2}\n]\nApproximate values:\n[\ny_1 \approx \frac{3 - 3.16}{2} = \frac{-0.16}{2} \approx -0.08, \quad y_2 \approx \frac{3 + 3.16}{2} = \frac{6.16}{2} \approx 3.08\n]", "---", "## Step 2: Analyze the Parabola’s Direction", "The quadratic coefficient ( a = 4 ) is positive, so the parabola opens upward. This means the expression 4y² - 12y - 1 is:\n- Negative between the two roots\n- Zero at the roots\n- Positive outside the roots", "---", "## Step 3: Determine the Solution to the Inequality", "Since we want 4y² - 12y - 1 ≤ 0, we seek where the expression is negative or zero — precisely between and including the roots.", "Thus, the solution is:\n[\ny \in \left[ \frac{3 - \sqrt{10}}{2}, \frac{3 + \sqrt{10}}{2} \right]\n]", "---", "## Step 4: Expressing the Final Answer Clearly", "For clarity and precision, the solution in interval notation is:\n[\n\boxed{ \frac{3 - \sqrt{10}}{2} \leq y \leq \frac{3 + \sqrt{10}}{2} }\n]", "In set notation:\n[\ny \in \left[ \frac{3 - \sqrt{10}}{2},\ \frac{3 + \sqrt{10}}{2} \right]\n]", "---", "## Why This Inequality Matters", "Understanding intervals where quadratic expressions are negative is vital in many areas, including:\n- Physics: modeling motion under constraints\n- Economics: profit and cost boundary analysis\n- Engineering: stability and stress testing\n- Computer Science: algorithm bounds and error margins", "Mastering techniques like identifying roots, analyzing parabola direction, and interpreting inequality signs prepares you for advanced math and real-life problem solving.", "---", "## Summary of Key Steps", "1. Write the equation: 4y² - 12y - 1 = 0\n2. Use the quadratic formula to find roots:\n [\n y_{1,2} = \frac{3 \pm \sqrt{10}}{2}\n ]\n3. Identify the parabola opens upward (a > 0).\n4. The expression ≤ 0 between the roots due to the shape of the parabola.\n5. Final solution:\n[\n\frac{3 - \sqrt{10}}{2} \leq y \leq \frac{3 + \sqrt{10}}{2}\n]", "---", "## Additional Tips", "- Always sketch the graph conceptually (a upward-opening parabola crossing the x-axis at two points) to visualize the solution.\n- Use calculators with symbolic computation for accuracy in values like (\sqrt{10}).\n- Practice transforming inequalities to standard form whenever tackling harder problems.", "---", "Mastering quadratics opens doors to deeper mathematical understanding. With consistent practice, solving inequalities like 4y² - 12y - 1 ≤ 0 becomes second nature — a powerful toolkit for both exams and real-world challenges.", "---", "Keywords for SEO:\nquadratic inequality 4y² - 12y - 1 ≤ 0, solve 4y² - 12y - 1 ≤ 0, quadratic equation roots, parabola inequality solver, real number solutions y, algebra step-by-step, how to solve 4y² - 12y - 1 ≤ 0, quadratic roots explanation, inequality solution guide, interpreting y-values from quadratic inequalities.", "---", "Start solving inequalities with confidence — your next math problem awaits!"]