Lens 06 of 9
The symmetry
Look at what stays the same when the variables swap or invert, or when the figure is turned.
An expression symmetric in x and 1/x is built from x + 1/x, so you never need x itself. Four quarter-circles in the corners of a square are one whole circle. Symmetry halves the work, or removes it: whatever is repeated can be counted once and multiplied.
The clues that call for it
- Quadratic Equations
An expression in x and 1/x
x + 1/x = 3 → square it: x² + 1/x² = 9 − 2 = 7.
- Areas Related to Circles
The same shape repeated in each corner
Four quadrants of radius r make one circle of area πr².
Worked example
If x + 1/x = 3, find x² + 1/x².
- 1
Don't solve for x
The target is symmetric in x and 1/x, so it is built from what you already have.
- 2
Square the given
The cross terms collapse to a constant.
(x + 1/x)² = x² + 2 + 1/x² = 9
- 3
Answer
x² + 1/x² = 7, and x itself was never needed.
9 − 2 = 7
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