This is a quadratic inequality. First, find the roots:
["# Solving a Quadratic Inequality: Finding the Roots First", "Understanding quadratic inequalities is a fundamental skill in algebra, and solving them begins with correctly identifying the roots of the associated quadratic equation. In this article, we’ll walk through the essential steps to find the roots of a quadratic inequality and explain why this foundation is crucial to solving the inequality effectively.", "## What Is a Quadratic Inequality?", "A quadratic inequality involves a quadratic expression—a polynomial of degree 2—on one side of the inequality sign (usually <, >, ≤, or ≥). For example:", "[\nax^2 + bx + c < 0\n]", "To solve such inequalities, the first and most critical step is determining the roots of the corresponding quadratic equation:", "[\nax^2 + bx + c = 0\n]", "These roots divide the number line into intervals, helping us test where the inequality holds true.", "---", "## Step 1: Find the Roots of the Quadratic Equation", "### Example Problem\nConsider the inequality:\n[\nx^2 - 5x + 6 < 0\n]", "The related quadratic equation is:\n[\nx^2 - 5x + 6 = 0\n]", "### Factoring the Quadratic\nTo find the roots, factor the quadratic expression:", "[\nx^2 - 5x + 6 = (x - 2)(x - 3)\n]", "Setting each factor equal to zero gives:", "[\nx - 2 = 0 \quad \Rightarrow \quad x = 2\n]\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "### Roots\nThe roots of the equation are:\n[\nx = 2 \quad \ ext{and} \quad x = 3\n]", "---", "## Why Are the Roots Important?", "The roots serve as key marker points on the number line that divide it into intervals. These intervals determine where the quadratic expression changes sign — from positive to negative or vice versa. For the example above, the number line is split into:", "- ( (-\infty, 2) )\n- ( (2, 3) )\n- ( (3, \infty) )", "By testing a point from each interval, we can determine where the inequality holds true.", "To summarize:\nRoots are the foundation for solving quadratic inequalities. Without finding them, you cannot determine the intervals to test or interpret the solution correctly.", "---", "## Next Steps: Solving the Inequality", "After finding the roots, follow these steps:", "1. Identify the roots ( x = 2 ) and ( x = 3 ).\n2. Divide the number line into intervals:\n - ( (-\infty, 2) )\n - ( (2, 3) )\n - ( (3, \infty) )\n3. Test a value in each interval in the original inequality ( x^2 - 5x + 6 < 0 ).\n4. Determine which intervals satisfy the inequality.", "In this case, the quadratic ( (x-2)(x-3) ) is negative (less than zero) only between the roots:", "[\n2 < x < 3\n]", "---", "## Conclusion", "Before solving any quadratic inequality, finding the roots of the associated quadratic equation is essential. These roots partition the number line into testing intervals that reveal where the inequality holds. Mastering this step ensures accurate interpretation and solution of quadratic inequalities — a cornerstone of algebra for students and problem solvers alike.", "Key Takeaways:\n- Roots are found by solving ( ax^2 + bx + c = 0 )\n- Roots divide the number line into intervals\n- Testing intervals determines where the inequality is true\n- Use roots as a foundation before solving the inequality", "Start strong by practicing root-finding — your path to confidently solving quadratic inequalities!", "---", "Keywords: quadratic inequality, find roots quadratic equation, solve quadratic inequality, roots of quadratic, algebraic methods, inequality solving steps, number line intervals, factoring quadratics, algebra tutorial.", "---", "Need help solving a specific quadratic inequality? Use the root-finding method described above — it’s the key to unlocking the solution!"]
