Elegant Substitution for Polynomial Equations.

Using symmetry and substitution to simplify algebra

Read it here if you’re not a Medium member. If this helped you see mathematics differently, support my writing ☕ here

Most difficult algebra problems aren’t difficult because the calculations are hard. They’re difficult because they invite you to look at them the wrong way.

Take the equation

On the surface, it seems like this is just an expansion problem, multiply out all of the terms and solve the resulting polynomial equation.

This is why it’s such a dangerous problem.

The traditional approach is to form a very large polynomial that seems to have little pattern to it at all. The clever approach starts by asking the following question

Why were these particular quadratics picked ?

When an algebraic equation looks especially difficult to compute, there is usually some deeper reason. This isn’t about computation speed ; it’s about recognizing the hidden message behind the equation.

As soon as you start seeing the two quadratics as the same object, you’ve cracked the problem open.

I Solved One Puzzle a Day for 100 Days . Here’s What It Did to My Brain

Take a moment. Take out your pen, paper, and other necessary supplies. First, give it a try. When you’re ready, continue on the path below

The Path to a Solution ➡️

Notice what happens when we complete the square.

Now the equation becomes

This changes how we see the problem.

Both factors share the same structure

The two expressions rarely are coincidence. The two expressions represent the same thing just in slightly different contexts ; algebra rewards us when we move to the center in each expression.

This substitution places both quadratic expressions symmetrically around zero.

This is not obviously so much easier at first sight. However, symmetry has started to arise.

Doesn’t look that much easier initially. Symmetry is now appearing however.

Whenever algebra presents a pair of expressions that differ only by a sign, there is usually an identity waiting to be used.

What matters is the relationship between them.

This is where many students instinctively expand everything. But expansion should be the last resort, not the first.

Instead, use the identity

Substituting the known values,

Working through the resulting algebra transforms the original equation into

This is the pivotal moment.

The problem has been reorganized into a familiar form.

Then factoring yields

giving the four solutions

A quick substitution confirms that all four satisfy the original equation.

The interesting part, however, isn’t the answer.

It’s the method.

Never approach a tough equation head-on ; always ask yourself

1. How similar are the components within this expression ?
2. Is there an underlying symmetry to be found ?
3. Could a new variable express this repetition ?
4. What would happen if I changed my point of view instead ?

These queries will probably hold greater importance than whatever algebraic trick you might learn.

The equation will likely be unique; the mindset, not.

And, as in so many things, in mathematics too, advancement sometimes comes not from doing more with your current approach, but from having a different one.✨…

Visit Medium to see this fascinating collection of challenges, questions, quizzes, puzzles, and tricks

[embed]List: 100 Days of Mathematical Puzzles 🧩🧠 | Curated by Thanushan Sivapatham | Medium 100 Days of Mathematical Puzzles 🧩🧠 · 97 stories on Mediumthanushansiva100.medium.com

Thank you for reading it! If you enjoy this puzzle, please remember to subscribe to get updates on my most recent articles, follow me for more, and give it a few claps. 😊

Got any questions or ideas ? Drop them in the comments below 💬 I’d love to hear from you! 🤗