Algebraic Symmetry Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
If x+y=10x + y = 10 and xy=21xy = 21, find x2+y2x^2 + y^2.

Solution

  1. 1
    Use symmetry: x2+y2=(x+y)2−2xy=100−42=58x^2 + y^2 = (x+y)^2 - 2xy = 100 - 42 = 58.
  2. 2
    No need to find individual values of xx and yy.

Answer

5858
Symmetric expressions like x2+y2x^2 + y^2 can be computed from the elementary symmetric polynomials x+yx + y and xyxy without solving for individual values. This is a powerful technique.

About Algebraic Symmetry

The property of an expression or equation that remains unchanged when certain transformations — such as swapping variables — are applied.

Learn more about Algebraic Symmetry →

More Algebraic Symmetry Examples

Example 1 easy

Is [formula] symmetric in [formula] and [formula]?

Example 2 medium

Is [formula] symmetric?

Example 3 easy

Is [formula] symmetric?

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