Solving algebraic equations rationally

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Aim:
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Finding and mastering new rational ideas for solving algebraic equations.

Example 1. Solve the equation

Solution

Example 2. Solve the equation .

Solution

i.e. the number 2 is represented as the sum of 1+1.

Example 3. Solve the equation

Solution If we use the standard method - eliminating the denominator and doing subsequent transformations we get the equation . We find the divisors of the free term, they are 32 numbers and through direct verification the roots of the equation are found.

The problem can be solved more rationally if the person solving it guesses that here the number 3 is suitable to be represented as 4-1.

Then the given equation can be presented in the form

The roots of the given equation are

Problem
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1.
** Solve the equation .

*Answer* .

Problem
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2.
** Solve the equation .

*Answer* .

By Rumyana Mavrova, Plovdiv university, rummav@pu.acad.bg