Final answer:
This high school mathematics problem is solved using algebra to create and solve a quadratic equation. The current ages of Mrs. Tandon's sons are determined by setting up equations based on the conditions provided and then applying the quadratic formula.
Step-by-step explanation:
The problem we are working on is a classic algebra question that involves creating and solving a quadratic equation to find the present ages of Mrs. Tandon's sons. To solve the problem, let's denote the age of the younger son as x years. Consequently, the age of the elder son becomes x+1 years since he is exactly one year older than the younger son. According to the problem, Mrs. Tandon's current age is the sum of the squares of her sons' ages. If we denote Mrs. Tandon's current age as T, then we have an equation:
T = x² + (x+1)²
Four years from now, Mrs. Tandon's age will be 5 times the age of her elder son. We can represent this with another equation:
T + 4 = 5(x + 1 + 4)
Solving these two equations, we can form a quadratic equation where the constant terms are derived from the given information. The general form of a quadratic equation is at² + bt + c = 0 and it can be solved using the quadratic formula which states that for any quadratic equation of the form ax² + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b² - 4ac)) / (2a)
Applying this to the equations we've formed for Mrs. Tandon and her sons, we can find the elder son's age first and subsequently deduce the younger son's age, thus solving the problem.