Section 3.5: Inequalities Involving Quadratic Functions

Solving Inequalities Involving a Quadratic Function

Quadratic inequalities involve expressions of the form ax^2 + bx + c > 0, ax^2 + bx + c < 0, ax^2 + bx + c \geq 0, or ax^2 + bx + c \leq 0, where a is nonzero. Solving these inequalities requires analyzing the corresponding quadratic function and determining where its graph lies above or below the x-axis.

• Step 1: Move all terms to one side so the other side is zero. The inequality should be in the form ax^2 + bx + c \gtrless 0.

• Step 2: Graph the quadratic function f(x) = ax^2 + bx + c using the vertex, y-intercept, and x-intercepts (if any).

• Step 3: Determine where the function is above the x-axis (for > or \geq) or below the x-axis (for < or \leq). This means finding the intervals where f(x) > 0 or f(x) < 0.

• Step 4: For non-strict inequalities (\geq or \leq), include the x-intercepts (solutions to ax^2 + bx + c = 0) in the solution set. For strict inequalities (> or <), exclude them.

Note: The solution set is typically expressed as intervals of x-values.

Example 1: Solving a Quadratic Inequality

Problem: Solve the inequality x^2 - 4x > 0 and graph the solution set.

• Rewrite as x^2 - 4x > 0.

• Factor: x(x - 4) > 0.

• Find zeros: x = 0 and x = 4.

• Test intervals: (-∞, 0), (0, 4), (4, ∞).

• Solution: x < 0 or x > 4.

Additional info: The graph of y = x^2 - 4x is a parabola opening upward, crossing the x-axis at x = 0 and x = 4. The solution set corresponds to the regions where the parabola is above the x-axis.

Example 2: Solving a Quadratic Inequality with Non-Strict Inequality

Problem: Solve the inequality x^2 + 4x + 3 \geq 0 and graph the solution set.

• Rewrite as x^2 + 4x + 3 \geq 0.

• Factor: (x + 1)(x + 3) \geq 0.

• Find zeros: x = -1 and x = -3.

• Test intervals: (-∞, -3], [-3, -1], [-1, ∞).

• Solution: x \leq -3 or x \geq -1.

Additional info: The solution includes the points where the graph touches the x-axis (x = -3 and x = -1) because the inequality is non-strict (\geq).

Example 3: Solving a Quadratic Inequality with Less Than or Equal To

Problem: Solve the inequality x^2 - x + 1 \leq 0 and graph the solution set.

• Rewrite as x^2 - x + 1 \leq 0.

• Attempt to factor or use the quadratic formula to find zeros.

• Discriminant: (-1)^2 - 4(1)(1) = 1 - 4 = -3 (no real roots).

• Since the parabola opens upward and does not cross the x-axis, f(x) > 0 for all x. Thus, there is no solution to the inequality.

Additional info: If the quadratic has no real roots and opens upward, it is always above the x-axis; if it opens downward, it is always below the x-axis.

Summary Table: Steps for Solving Quadratic Inequalities