Question 1: Factor the quadratic expression x² + 6x and express it in completed square form.

Answer: (x + 3)²

Question 2: Expand the expression (x + 5)² and write it in the form x² + bx.

Answer: x² + 10x + 25

Question 3: Calculate the value of k such that x² + 4x + k is a perfect square.

Answer: 4

Question 4: Rewrite the quadratic x² - 10x + 16 in completed square form.

Answer: (x - 5)² - 9

Question 5: Explain how to complete the square for the expression x² + 8x + 3. What steps are involved?

Answer: To complete the square, take half of 8, which is 4, and square it to get 16. Rewrite as x² + 8x + 16 - 16 + 3 = (x + 4)² - 13.

Question 6: Solve for x: x² + 6x = 0 by completing the square.

Answer: x = 0 or x = -6

Question 7: A rectangle has width x and length x + 4. Write an expression for its area in x² + bx form and complete the square to find the dimensions when the area is 100.

Answer: Area = x(x + 4) = x² + 4x. Completing the square: x² + 4x + 4 - 4 = (x + 2)² - 4. Set equal to 100: (x + 2)² - 4 = 100, so (x + 2)² = 104, x + 2 = ±√104.

Question 8: Construct a quadratic in the form x² + bx that has roots at x = 1 and x = -3.

Answer: x² + 2x - 3

Question 9: Identify the error in the following attempt to complete the square: x² + 2x + 1 = (x + 1)² + 2.

Answer: The error is adding 2 instead of subtracting it; correct expression is (x + 1)².

Question 10: Challenge: Find all solutions to the equation x² + 10x + 25 = 0 by completing the square.

Answer: x = -5 (double root)