Create a quadratic inequality that represents the following phrase:
The square of the quantity 5 subtracted from x has a minimum value of 50.
The quadratic inequality that represents the given phrase is:
(x - 5)^2 ≥ 50
Explanation:
The given phrase can be written as:
(x - 5)^2 = (x - 5)(x - 5) = x^2 - 10x + 25
To find the minimum value of this expression, we can use the fact that the vertex of a quadratic function in the form of ax^2 + bx + c is located at x = -b/2a.
In this case, a = 1, b = -10, and c = 25. So the x-coordinate of the vertex is:
x = -(-10) / 2(1) = 5
Therefore, the minimum value of the expression is:
x^2 - 10x + 25 = (5)^2 - 10(5) + 25 = 0
To ensure that the expression is greater than or equal to 50, we can write the quadratic inequality:
(x - 5)^2 ≥ 50