Solving algebraic equations rationally

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

Example 1. Solve the equation tema4_EN_files\tema4_EN_MathML_0.jpg




Example 2. Solve the equation tema4_EN_files\tema4_EN_MathML_3.jpg.


tema4_EN_files\tema4_EN_MathML_4.jpg i.e. the number 2 is represented as the sum of 1+1.


Example 3. Solve the equation tema4_EN_files\tema4_EN_MathML_6.jpg

Solution If we use the standard method - eliminating the denominator tema4_EN_files\tema4_EN_MathML_7.jpg and doing subsequent transformations we get the equation tema4_EN_files\tema4_EN_MathML_8.jpg. 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 tema4_EN_files\tema4_EN_MathML_9.jpgtema4_EN_files\tema4_EN_MathML_10.jpg

The roots of the given equation are tema4_EN_files\tema4_EN_MathML_11.jpg

For individual work

Problem  1. Solve the equation tema4_EN_files\tema4_EN_MathML_12.jpg.

Answer tema4_EN_files\tema4_EN_MathML_13.jpg.

Problem  2. Solve the equation tema4_EN_files\tema4_EN_MathML_14.jpg.

Answer tema4_EN_files\tema4_EN_MathML_15.jpg.

By Rumyana Mavrova, Plovdiv university,