Challenge Level

The following equations are difficult to solve by direct attack but if you look for symmetric features and make simple substitutions they become much easier to solve.

(1) $\frac{2x}{2x^2-5x+3}+\frac{13x}{2x^2+x+3}=6.$

(2) $x^4 -8x^3 + 17x^2 -8x + 1 = 0.$

(3) $(x-4)(x-5)(x-6)(x-7) = 1680.$

(4) $(8x+7)^2(4x+3)(x+1)=\frac{9}{2}.$

NOTES AND BACKGROUND

This is an example of a process which occurs frequently in mathematics. Let's refer to two frames of reference as A and B and say we have a problem stated in A, then the technique is to map the given relationships to B, work in B and then map the results back to A. All these equations have symmetry of one sort or another. By using the symmetry to make a substitution you can change the variable and get an equation which is easier to solve. After that you have to use the solutions you have found and go back to find the corresponding solutions of the original equation.

To find out more about this general technique see the article "The Why and How of Substitution".