Apply the quadratic formula to a sorted array such that the resulting array is still sorted.

This question is combining concepts from both mathematics (specifically, quadratic equations) and computer science (algorithms and sorting). It's asking for you to apply the quadratic formula to each element in a sorted array such that the resulting array remains sorted. This implies that the quadratic formula we're considering has at least one positive real root.

Let's break down the question:

1. The quadratic formula, usually expressed as (-b ± sqrt(b^2 - 4ac)) / (2a), is used to find the roots of a quadratic equation ax^2 + bx + c = 0.

2. An Array in programming usually is a static set of elements of the same type, sorted or unsorted.

3. Sorted array: Sorted arrays means the elements are in sequential order. This can be either ascending or descending depending on the context.

The key points to consider answering this question:

1. Understanding of quadratic formula and its effect on numbers (positive and negatives).
2. Understanding of sorting in programming— preserving the order after altering elements.
3. Choosing an algorithmic approach to efficiently perform this task.

Here is a simple way to apply the quadratic formula to a sorted array:

Given a sorted array as input: arr[] = {1, 2, 3, 4, 5}

Firstly, apply the quadratic formula to each element in the array. 

We will consider a simple quadratic equation: x^2 + x + 1 for this example, so the application of the quadratic formula to each element may look like this:

-1 ± √(1 - 4(1-x)) / 2(1)

After applying the quadratic formula, new values of arr[] can be {-0.5, 0.5, 1.5, 2.5, 3.5}

Even after the quadratic transformation, as you can observe the array is still sorted.

One thing you need to remind yourself is that Quadratic formulas can throw complex numbers which are imaginary numbers.

Given a sorted array as input: arr[] = {1, 2, 3, 4, 5}

Suppose your quadratic equation was x^2 + x - 4, then applying the quadratic formula would give:

((-1) ± √(1 - 4(-4 + x)))/2

When you input x=1, this simplifies to:

((-1) ± √(1 - 1)) / 2

As you can see, the part under the square root (known as the discriminant) is 0. This means the equation has one real and rational root.

Thus, when you apply the quadratic formula to this sorted array with this equation, the resulting array remains sorted but the transformation can give you complex numbers which could be handled separately.