Why do this problem?

Making substitutions to make the task in hand easier is an example of a valuable technique in many areas of mathematics (eg integration, transformations of functions, change of axes, diagonalisation of matrices etc.).  By introducing students to this technique as an example of a general process it will help them to understand what is going on when they meet the process in many different mathematical situations. These equations give students useful practice in algebraic manipulation. They will need to look for symmmetrical features in the expressions and exploit the symmetry to make it easier to solve the equations.

Possible approach 

This problem featured in the NRICH Secondary webinar in June 2022.

It is helpful to introduce this problem with some discussion about switching frames of reference to make the equations easier to solve and how you are looking for symmetry in order to choose a good substitution.

Start by asking how pupils might solve $(x+3)^2+(x+5)^2=6$.

Follow up by asking if they notice anything about the expressions in the brackets.  Can they relate them to another expression?

Introduce the idea of a substitution, such as $y=x+4$.  Ask how each of the brackets is related to $x+4$ and how we could rewrite the equation in terms of $y$.  Ask pupils to then solve the equation in terms if $y$.  How can we then find $x$?

Ask students which method they though was easiest - then ask them to solve $(x+3)^4+(x+5)^4=34$.

There are several other equations in this problem where symmetry can be used to find a helpful substitution.

Possible extension ideas
 

• $(2x - 3)^4 + (2x - 5)^4 = 2$

• $ (x - 2)(x + 1)(x + 4)(x + 7) = 19$

• $(12x - 1)(6x - 1)(4x - 1)(3x - 1) = 5$

Possible support

Make up your own equation by taking a simple equation and making a substitution to make it more complicated. For example take any quadratic equation in $x$ and turn it into a quartic equation by substituting $x=z+\frac{1}{z}.$