These are two questions and two answers.
Question 1. For the given quadratic equation convert into vertex form, find the vertex, and find the value for x = 6
y = -2x^2 + 2x + 2
Solution
1) The vertex form of a quadratic equation is y = A(x -h)^2 + k, where the vertex is (h,k)
2) To convert y = -2x^2 + 2x + 2 complete squares as shown below.
3) common factor -2 for the first two terms => y = -2 [x^2 - x] + 2
4) complete squares for x^2 - x => (x -1/2)^2 - 1/4
=> y = - 2 [ (x - 1/2)^2 - 1/4 ] + 2
5) take -1/4 out of the square brackets => y = - 2(x - 1/2)^2 + 1/2 + 2
=> y = -2 (x - 1/2)^2 + 5/2 this is the final vertex form
6) The vertex is (1/2, 5/2)
Answer: the vertex is (1/2, 5/2)
7) value for x = 6
y = -2(6 - 1/2)^2 + 5/2 = - 58
or y = -2(6)^2 + 2(6) + 2 = - 58
Answer: - 58
Question 2. box shaped like a cube with side length (5a + 4b). Volume ?
Solution:
1) Volume, V = (side length)^3
2) V = (5a + 4b)^3
3) Use cube a sum expansion: (x + y)^3 = x^3 + 3x^2 y + 3x y^2 + y^3
=> V = (5a)^3 + 3(5a)^2 (4b) + 3(5a)(4b)^2 + (4b)^3
=> V = 125a^3 + 3(25a^2) (4b) + 15a(16b^2) + 64b^3
=> V = 125a^3 + 300a^2 b + 240a b^2 + 64b^3
Answer: 125a^3 + 300a^2 b + 240a b^2 + 64b^3
