Answer:
The equation which reveals the minimum value for the equation is 2·x² + 12·x - 14 = 0 is the option;
2·(x + 3)² = 32
Explanation:
The given function can be presented as follows;
2·x² + 12·x - 14 = 0
The quadratic equation in standard form is a·(x - h)² + k
Where;
(h, k) is the vertex of the parabola
Given that the coefficient of the x² is positive, the graph is U-shaped and the lowest point is given by the vertex of the graph as follows;
2·x² + 12·x - 14 = 2·x² + 12·x - 14 = 2·(x² + 6·x - 7) = 0
2·(x² + 6·x - 7) = 2·(x² + 6·x + 9 - 7 - 9) = 2·((x + 3)² - 16)
2·((x + 3)² - 16) = 2·(x + 3)² - 32 = 0
∴ 2·x² + 12·x - 14 = 2·(x + 3)² - 32 in vertex form
The minimum value (vertex, (h, k)) for the equation = (-3, -32)
Therefore, the equation which reveals the minimum value for the equation is 2·(x + 3)² = 32