Question 1: Solve the quadratic inequality x^2 - 5x + 6 > 0 using a sign diagram.

Answer: x < 2 or x > 3

Question 2: Construct a sign diagram for the inequality x^2 + 4x + 3 ≤ 0 and state the solution set.

Answer: x = -1 or x = -3

Question 3: Determine the solution to the inequality (x - 2)(x + 1) < 0 by analyzing the sign diagram.

Answer: x between -1 and 2

Question 1: Given the quadratic inequality 3x^2 - 12x + 4 ≥ 0, draw its sign diagram and explain how it helps determine the solution set.

Answer: The roots are at x = (12 ± √(144 - 48))/6; sign diagram shows x ≤ root1 or x ≥ root2, solutions are outside roots.

Question 2: A parabola opens upward with roots at x = -2 and x = 3. Use a sign diagram to find where the quadratic is negative.

Answer: x between -2 and 3

Question 1: A ball is thrown upwards such that its height h (in meters) after t seconds is given by h = -5t^2 + 20t. Use a sign diagram to find when the ball is above 10 meters.

Answer: Solve -5t^2 + 20t > 10; roots at t=0 and t=4; height >10 between t=0 and t=4.

Question 1: Construct the sign diagram for the quadratic inequality 2x^2 - 3x - 2 < 0 and determine the solution set. Describe the steps involved.

Answer: Roots at x=2 and x=-0.5; parabola opens upward; inequality negative between roots.

Question 2: Given the quadratic 4x^2 + bx + 1 ≥ 0, find the range of b for which the quadratic is always non-negative. Use sign diagrams to justify your answer.

Answer: Discriminant ≤ 0; b^2 - 16 ≥ 0; b ≤ -4 or b ≥ 4

Question 1: Plot the graph of y = (x - 1)(x + 2) and identify the intervals where y is negative, zero, or positive using a sign diagram.

Answer: y negative between -2 and 1, zero at -2 and 1, positive outside.

Question 2: A quadratic function has roots at x = -3 and x = 4. If the quadratic opens downward, where is it positive? Draw the sign diagram and justify.

Answer: Quadratic positive outside roots (x < -3 or x > 4).

Question 1: A student draws a sign diagram for x^2 - 4x + 3 ≥ 0 but labels the regions incorrectly, thinking the inequality is < 0 everywhere. Identify the mistake and correct the solution.

Answer: Mistake: mislabeling the sign regions; correction: roots at x=1 and x=3, parabola opens upward, so ≥ 0 outside the roots.