Solve the quadratic inequality using the quadratic formula:

["# Solve the Quadratic Inequality Using the Quadratic Formula", "Quadratic inequalities are essential in algebra and real-world applications, from physics to economics. Understanding how to solve them is a crucial skill for students and math enthusiasts alike. In this article, we’ll explore how to solve quadratic inequalities using the quadratic formula, step-by-step instructions, and practical examples to help you master this important technique.", "---", "## What Is a Quadratic Inequality?", "A quadratic inequality is an inequality involving a quadratic expression — that is, a polynomial of degree 2, typically written in the form:", "[\nax^2 + bx + c < 0, \quad ax^2 + bx + c > 0, \quad \ ext{or} \quad ax^2 + bx + c = 0\n]", "Here, (a), (b), and (c) are real numbers, and (a <br/>\ne 0). Solving a quadratic inequality means finding all real values of (x) that satisfy the inequality.", "---", "## Why Use the Quadratic Formula?", "While factoring is one method for solving quadratic equations, it works only when the quadratic expression factors neatly into binomials. When factoring is difficult or impossible, the quadratic formula becomes a powerful tool. The formula gives the exact roots of any quadratic equation:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Knowing the roots allows us to determine the intervals on the number line where the quadratic expression is positive or negative.", "---", "## Step-by-Step: Solving Quadratic Inequalities Using the Quadratic Formula", "### Step 1: Write the Inequality in Standard Form", "Arrange the inequality so the quadratic expression is set less than 0, greater than 0, or equal to 0:", "[\nax^2 + bx + c < 0 \quad \ ext{(less than zero)}\n]\n[\nax^2 + bx + c > 0 \quad \ ext{(greater than zero)}\n]", "If the expression equals zero somewhere, include equality (e.g., ≥ or ≤) as needed.", "### Step 2: Use the Quadratic Formula to Find Roots", "Solve the related quadratic equation:", "[\nax^2 + bx + c = 0\n]", "Applying the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This gives two real roots, or one, or possibly no real roots if the discriminant (D = b^2 - 4ac) is negative.", "### Step 3: Analyze the Discriminant", "- If (D > 0): two distinct real roots → expression crosses the x-axis at two points.\n- If (D = 0): one repeated real root → touches the x-axis at one point.\n- If (D < 0): no real roots → expression never crosses the x-axis.", "This affects how we analyze intervals independently of sign.", "### Step 4: Determine the Sign of the Quadratic Expression", "The sign of (ax^2 + bx + c) depends on:\n- The leading coefficient (a): determines whether the parabola opens upward ((a > 0)) or downward ((a < 0)).\n- The roots found in Step 2.", "#### Case 1: Two distinct real roots ((D > 0))\nThe parabola crosses the x-axis at two points. The expression is:\n- Positive between the roots (if (a > 0)),\n- Negative outside the roots.", "#### Case 2: One real root ((D = 0))\nThe parabola touches the x-axis at one point. The expression is:\n- Negative on both sides (except at the root),\n- Zero at the root.", "#### Case 3: No real roots ((D < 0))\nThe parabola does not intersect the x-axis. The sign depends only on (a):\n- If (a > 0), the expression is always positive.\n- If (a < 0), the expression is always negative.", "---", "### Step 5: Express the Solution Set", "Use interval notation to write all (x) values satisfying the inequality.", "---", "## Examples", "### Example 1: Solve (x^2 - 5x + 6 < 0)", "- Step 1: Already standard form.\n- Step 2: Use quadratic formula:\n (a = 1), (b = -5), (c = 6)\n [\n x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}\n ]\n Roots: (x = 3) and (x = 2)\n- Step 3: (D = 1 > 0), two real roots. Parabola opens upward ((a = 1 > 0)).\n- Step 4: Since parabola opens up, expression is negative between the roots.\n- Step 5: Solution:\n [\n 2 < x < 3\n ]\n In interval notation: ((2, 3))", "---", "### Example 2: Solve (x^2 + 4x + 4 > 0)", "- Step 1: Standard form.\n- Step 2:\n [\n x = \frac{-4 \pm \sqrt{16 - 16}}{2} = \frac{-4}{2} = -2\n ]\n One repeated root.\n- Step 3: (D = 0), parabola touches x-axis at (x = -2), opens upward ((a = 1 > 0)).\n- Step 4: Positive everywhere except at (x = -2), where expression is zero.\n- Step 5: Solution:\n [\n x <br/>\ne -2\n ]\n In interval notation: ((-\infty, -2) \cup (-2, \infty))", "---", "## Tips for Success", "- Always simplify the inequality into standard form first.\n- Carefully compute the discriminant (D = b^2 - 4ac) to know how many real roots exist.\n- Pay attention to the inequality symbol: <, >, ≤, or ≥ affects whether endpoints are included.\n- Always test intervals or plot key points to confirm the sign.\n- Use a number line to visualize the solution.", "---", "## Summary", "Solving quadratic inequalities using the quadratic formula is a reliable method that works for all quadratics. By finding the roots, analyzing the parabola’s direction and position, and testing intervals, you can clearly determine where the expression satisfies the inequality. Mastering this technique builds a strong foundation for working with polynomials and real-world modeling.", "---", "### Key Takeaways", "- Use the quadratic formula when factoring is not obvious.\n- Roots divide the number line into intervals.\n- Sign of (a) and number of real roots determine where the expression is positive or negative.\n- Combine root knowledge with inequality direction for accurate solutions.", "Start practicing with different quadratics to build confidence — soon, solving inequalities will become second nature!", "---", "Keywords: quadratic inequality, quadratic formula, solve quadratics, algebra 2, work with inequalities, quadratic equation solutions, factoring with quadratic formula, real roots, number line graph, of inequalities."]