Question 1: Solve for the critical values of the quadratic equation x^2 - 4x - 5 = 0.

Answer: 5 and -1

Question 2: Find the critical values of 2x^2 + 3x - 2 = 0.

Answer: -1 and 0.333...

Question 3: Determine the solutions where x^2 + 6x + 9 = 0.

Answer: -3 (repeated root)

Question 1: Given the quadratic inequality x^2 - 3x - 4 > 0, find and interpret the critical values, then determine the solution set.

Answer: Critical values are x=4 and x=-1; solution: x < -1 or x > 4

Question 2: Explain why the critical values of x^2 - 2x - 8 = 0 are important when solving x^2 - 2x - 8 < 0.

Answer: Critical values are the roots, x = 4 and x = -2; solutions are between these roots, i.e., -2 < x < 4, where the parabola is below the x-axis.

Question 3: Construct the intervals where the quadratic x^2 + 5x + 6 > 0 based on its critical values.

Answer: Critical values x = -2 and x = -3; solution: x < -3 or x > -2

Question 1: A ball is thrown so that its height h in meters after t seconds is given by h = -4.9t^2 + 20t + 1. Find the critical points to determine when the ball reaches its maximum height.

Answer: Critical point at t = 2.04 seconds; maximum height at this time.

Question 1: For the quadratic inequality 3x^2 - 7x + 2 < 0, find the critical values and analyze the solution set. Then, discuss potential misconceptions students might have about the solution interval.

Answer: Critical values x=0.5 and x=2.67; solution: 0.5 < x < 2.67; misconception: confusing the roots with the intervals where the parabola is above/below x-axis.

Question 2: Derive the critical values for the quadratic function f(x) = -2x^2 + 8x - 6 and explain their significance in the context of the inequality f(x) > 0.

Answer: Critical value at x=2; at this point, f(x) reaches maximum; solutions where f(x)>0 are x<1 and x>3.

Question 1: A student solves x^2 - 5x + 6 = 0 and states the critical values are x=5 and x=6. Identify the mistake and provide the correct critical values.

Answer: Mistake: incorrect roots. Correct critical values are x=2 and x=3.

Question 2: A student claims that the critical values of x^2 + 4x + 4 = 0 are x=4 and x=-4. Explain what is wrong and list the actual critical value.

Answer: Error: misunderstanding roots. Actual critical value is x=-2 (double root).