Final answer:
The student's equation is solved by rearranging it into a quadratic form and applying the quadratic formula to find the real root, while ignoring any imaginary solutions.
Step-by-step explanation:
The student is asking to solve the following equation for the real root: ((1.000+2.000x)²)/(1.000-x)=1.110. To find the solution, we manipulate the equation so that it can be rearranged into a quadratic equation in the form ax² + bx + c = 0. This involves expanding the expression, simplifying, and then applying the quadratic formula where necessary.
Focusing on the hint that the equation resembles a perfect square, we understand that like 5¹ · 5¹, we can represent a square root with a fractional power, simplifying the equation further.
Using the quadratic formula, we solve for x to find its real roots, ignoring any imaginary components. In such problems, we evaluate the solutions for both signs in the numerator, considering the physical or realistic constraints, as some solutions may not be valid in real-world scenarios.