Question 1: Solve the quadratic inequality x^2 - 4x - 5 > 0 by constructing its sign diagram.

Answer: x < -1 or x > 5

Question 2: Determine the intervals where the quadratic y = x^2 - 3x + 2 is less than zero using a sign diagram.

Answer: x between 0 and 2

Question 3: Plot the graph of y = -x^2 + 4x - 3 and identify the intervals where y is positive. Draw the sign diagram to support your answer.

Answer: x between 1 and 3

Question 4: Construct a sign diagram for the inequality 2x^2 - 8x + 6 ≥ 0 and interpret the solution set.

Answer: x ≤ 1 or x ≥ 3

Question 5: Identify the error in the following sign diagram for x^2 - 5x + 6 < 0 and correct it.

Answer: The sign diagram incorrectly shows the intervals; the correct intervals are between 2 and 3.

Question 6: Find the solution to the inequality x^2 + 6x + 9 < 0. Use a sign diagram to explain.

Answer: No solution, since x^2 + 6x + 9 = (x+3)^2 ≥ 0 always

Question 7: Given y = x^2 - 2x - 8, draw its sign diagram and identify where y > 0.

Answer: x < -2 or x > 4

Question 8: A real-world scenario: A garden bed is longer than 4 meters but shorter than 10 meters. Write an inequality to represent this and construct its sign diagram.

Answer: x > 4 and x < 10

Question 9: Construct a sign diagram for 3x^2 - 12x + 12 ≥ 0 and solve the inequality.

Answer: x ≤ 2 or x ≥ 4

Question 10: Explain why the quadratic y = x^2 + 4x + 5 does not have a sign diagram showing intervals where it is less than zero.

Answer: Because its discriminant is negative, so y is always positive.

Question 11: Determine the solution set for x^2 - 7x + 10 ≤ 0 by constructing the sign diagram.

Answer: x between 2 and 5

Question 12: Challenge: The quadratic 2x^2 + 3x - 2 has a sign diagram with roots at x = -1 and x = 0.5. Sketch the sign diagram and determine where the quadratic is negative.

Answer: x between -1 and 0.5