3^2 - 4(y - 2)(y - 1) \geq 0
["SEO-Optimized Guide: Solving the Inequality 3² - 4(y - 2)(y - 1) ≥ 0", "---", "# Solving the Inequality ( 3^2 - 4(y - 2)(y - 1) \geq 0 ): Step-by-Step Explanation", "Understanding how to solve quadratic inequalities is essential for students in algebra and anyone working with polynomial expressions. One common form students encounter is inequalities involving expressions like ( 3^2 - 4(y - 2)(y - 1) \geq 0 ). In this guide, we break down how to simplify and solve this inequality step-by-step, along with clear explanations optimized for search engines (SEO).", "---", "## Step 1: Simplify the Inequality", "Start by simplifying the expression:", "[\n3^2 - 4(y - 2)(y - 1) \geq 0\n]", "Since ( 3^2 = 9 ), the inequality becomes:", "[\n9 - 4(y - 2)(y - 1) \geq 0\n]", "The next focus is on expanding the quadratic term ( (y - 2)(y - 1) ).", "---", "## Step 2: Expand ( (y - 2)(y - 1) )", "Use the distributive property (FOIL method):", "[\n(y - 2)(y - 1) = y^2 - y - 2y + 2 = y^2 - 3y + 2\n]", "Substitute back into the inequality:", "[\n9 - 4(y^2 - 3y + 2) \geq 0\n]", "Now distribute the (-4):", "[\n9 - 4y^2 + 12y - 8 \geq 0\n]", "Combine like terms:", "[\n(-4y^2 + 12y + 1) \geq 0\n]", "So the inequality simplifies to:", "[\n-4y^2 + 12y + 1 \geq 0\n]", "---", "## Step 3: Multiply by –1 (and Reverse Inequality)", "To make the quadratic coefficient positive (easier to analyze), multiply both sides by (-1). Remember: multiplying or dividing an inequality by a negative number reverses the inequality sign.", "[\n4y^2 - 12y - 1 \leq 0\n]", "---", "## Step 4: Solve the Quadratic Equation ( 4y^2 - 12y - 1 = 0 )", "Use the quadratic formula:", "[\ny = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 4 ), ( b = -12 ), ( c = -1 ). Plug in the values:", "[\ny = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(-1)}}{2(4)} = \frac{12 \pm \sqrt{144 + 16}}{8} = \frac{12 \pm \sqrt{160}}{8}\n]", "Simplify ( \sqrt{160} = \sqrt{16 \cdot 10} = 4\sqrt{10} ):", "[\ny = \frac{12 \pm 4\sqrt{10}}{8} = \frac{4(3 \pm \sqrt{10})}{8} = \frac{3 \pm \sqrt{10}}{2}\n]", "So the roots are:", "[\ny_1 = \frac{3 - \sqrt{10}}{2}, \quad y_2 = \frac{3 + \sqrt{10}}{2}\n]", "---", "## Step 5: Determine the Interval Where the Inequality Holds", "The quadratic expression ( 4y^2 - 12y - 1 \leq 0 ) holds between the two real roots because the parabola opens upwards (coefficient of ( y^2 ) is positive).", "Thus, the solution is:", "[\n\frac{3 - \sqrt{10}}{2} \leq y \leq \frac{3 + \sqrt{10}}{2}\n]", "---", "## Step 6: Final Answer in Boxed Format", "[\n\boxed{ \frac{3 - \sqrt{10}}{2} \leq y \leq \frac{3 + \sqrt{10}}{2} }\n]", "This interval represents all real values of ( y ) satisfying ( 3^2 - 4(y - 2)(y - 1) \geq 0 ).", "---", "## SEO Keywords Summary\n- Solve quadratic inequality\n- ( 3^2 - 4(y - 2)(y - 1) \geq 0 )\n- Simplify quadratic expressions\n- Solve ( 4y^2 - 12y - 1 \leq 0 )\n- Step-by-step algebra\n- High school algebra tips\n- Inequality solution guide", "---", "## Why This Matters for Students and Educators", "Solving inequalities like ( 3^2 - 4(y - 2)(y - 1) \geq 0 ) strengthens understanding of quadratic functions, domain analysis, and function behavior. Mastering these techniques prepares learners for higher-level math, SAT/ACT problem-solving, and real-world data modeling.", "---", "Explore related topics:\n- Graphing quadratic inequalities\n- Finding zeros of quadratic functions\n- Applications of quadratic inequalities in physics and economics", "---", "Keep practicing to master the logic behind inequalities — clarity comes with repetition and precise algebraic steps!"]
