Question 1: Rewrite the quadratic expression x² + 6x in vertex form by completing the square.

Answer: (x + 3)² - 9

Question 2: Identify the value of b in the expression x² + bx if the expression is equivalent to (x + 4)² - 9.

Answer: 8

Question 3: Complete the square for x² - 10x + 21 and write the expression in vertex form.

Answer: (x - 5)² - 4

Question 4: Simplify the expression x² + 8x + 15 and express it in completed square form.

Answer: (x + 4)² - 1

Question 1: A ball is thrown such that its height h (in meters) after t seconds can be modeled by h = -4t² + 8t + 2. Rewrite the height function in the form of x² + bx to identify the maximum height.

Answer: h = -4(t² - 2t - 0.5) ; in x² + bx form: h = -4(x² - 2x + 1/4) + 3; maximum height is 3 meters

Question 2: Explain why completing the square helps in finding the vertex of a quadratic expressed in x² + bx form.

Answer: Completing the square rewrites the quadratic as a perfect square trinomial plus a constant, making the vertex (minimum or maximum point) easily identifiable from the squared term.

Question 3: Determine the vertex of the quadratic y = x² + 4x + 1 by completing the square.

Answer: Vertex at (-2, -3)

Question 4: A parabola opens downward and has its vertex at (2, 5). Write a possible quadratic expression in x² + bx form.

Answer: -a(x - 2)² + 5

Question 1: A rectangular garden's area A (in m²) is modeled by A = x² + 10x, where x is the length of one side in meters. Find the length x that maximizes the area.

Answer: Complete the square: A = (x + 5)² - 25; maximum area when x = -5 (not feasible for length); since length cannot be negative, the maximum area at x=0, area = 0.

Question 2: A company designs a satellite dish where the shape's profile is modeled by y = -x² + 4x + 1. Find the value of x that gives the maximum height of the dish.

Answer: x=2

Question 1: Given the quadratic expression x² + bx + c, and knowing it has roots at x=1 and x=4, find the values of b and c and write the quadratic in x² + bx form.

Answer: b = -5, c=4; quadratic: x² - 5x + 4

Question 2: Prove that rewriting x² + 8x + 16 as a perfect square provides the same minimum point as completing the square process.

Answer: x² + 8x + 16 = (x + 4)²; minimum at x = -4, which matches the completing the square method.

Question 1: A student writes x² + 12x as (x + 6)² - 36. Is this correct? If not, identify and correct the mistake.

Answer: Yes, the expansion is correct.

Question 2: A common misconception is that the coefficient b in x² + bx must always be positive for completing the square. Explain why this is incorrect.

Answer: Completing the square works regardless of whether b is positive or negative; the sign affects the process but not its validity.

Question 3: Rewrite the expression x² - 4x + 7 in vertex form. Show your working.

Answer: (x - 2)² + 3