Introduction
Understanding how to apply algebraic identities to solve for the sum of the squares of two numbers, given their sum and the sum of their cubes, is a valuable skill in mathematics. This problem involves the application of various algebraic concepts, including identities and simultaneous equations. In this article, we will explore a step-by-step solution to find the sum of the squares of two numbers, given that the sum of these numbers is 13 and the sum of their cubes is 793.
Understanding the Problem
We are given two numbers, x and y, such that:
The sum of the numbers is 13, i.e., x y 13.
The sum of the cubes of the numbers is 793, i.e., x^3 y^3 793.
The objective is to find the sum of their squares, x^2 y^2.
Solution
To solve this problem, we will use algebraic identities to manipulate the given equations and find the desired sum.
Step 1: Use the identity for the sum of cubes
We start by using the identity for the sum of cubes:
x^3 y^3 (x y)(x^2 - xy y^2)
Substituting the given values:
793 13(x^2 - xy y^2)
Simplifying, we get:
x^2 - xy y^2 frac{793}{13} 61
Step 2: Express x^2 - xy y^2 in terms of xy
Since we know that x y 13, we can square both sides to get:
(x y)^2 13^2 169
This can be expanded to:
x^2 2xy y^2 169
Now we can express x^2 - xy y^2 as:
x^2 - xy y^2 (x^2 2xy y^2) - 3xy 169 - 3xy
Substituting this into the equation from Step 1:
169 - 3xy 61
Solving for xy, we get:
3xy 108 Rightarrow xy 36
Step 3: Find the sum of the squares x^2 y^2
Since we know:
x^2 2xy y^2 169
We can substitute the value of xy to find x^2 y^2:
x^2 y^2 169 - 2xy 169 - 2(36) 169 - 72 97
Therefore, the sum of the squares of the two numbers is 97.
Conclusion
In this article, we have explored the solution to a complex algebraic problem by applying various identities and concepts. The sum of the squares of two numbers, given their sum and the sum of their cubes, can be found using a step-by-step approach that involves manipulating the given equations and using algebraic identities. The final result is that the sum of the squares of the two numbers is 97.
Understanding these concepts not only helps in solving specific problems but also in developing a deeper understanding of algebra and its applications. If you found this article helpful, please consider following us on social media and upvoting it for future reference.