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Only a Few People Can Solve This Math Problem
When I look at an algebra problem, I first ask whether the expressions have a relationship worth preserving. Expanding everything can make that relationship harder to see.
Consider the equation
It invites an immediate response: square both sides. That is a valid starting point, provided we keep track of signs. It gives
and hence
A short equation has become a fourth-degree polynomial. We can solve it, but a substitution makes the structure easier to understand.
Before doing any algebra, notice that a real square root is nonnegative. Therefore, any solution must satisfy
Together, these conditions reduce to
This restriction will help us distinguish genuine solutions from roots introduced by algebraic transformations.
Now give the square root a name:
The definition gives y²=5-x, while the original equation gives y=5-x². Rearranging, we obtain
Here is the useful symmetry: exchanging x and y exchanges the two equations.
Subtracting them removes the constant:
Factor the difference of squares, then take out the common factor:
