Today’s Challenge
I came across a very interesting problem on Twitter from @besdakikadamat1.
We are asked to find which expression is greater.
Give the problem a try before jumping in for the solution!
Solution
To start with, we begin with the observation that if a and b are positive and that
then, we have
The converse is also true as long as a and b are positive. With that in mind, the strategy is to square both expressions so that we can get rid of the square roots and evaluate them.
We will start work with both expressions simultaneously, using the formula for the perfect square, we have
Notice the highlighted parts, 2020 + 2023 = 2021 + 2022 = 4043, therefore we can cancel them on both sides.
Besides the 2 in front of the square roots can also be cancelled leaving us with
Now, recall our initial observation, we can essentially apply the rule once more by squaring both expressions, giving us
Ok! Stop scrolling and think about what you can do at this stage!
Hint: factorization!
I am assuming you have thought that through. Well to solve this analytically, we can indeed rewrite our expressions as
Upon expanding them, we arrive at
See? The expression on the right-hand side is greater than that on the left by 2!
Therefore, we can conclude that
Well done!
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Happy reading,
Barry 🍩