How To Really Solve This Tricky Algebra Problem? (V)

Welcome to the fifth entry in the tricky algebra problem series. Over the past couple of iterations, the difficulty level has been steadily rising. So, this time around, I wanted to make things a little beginner friendly and lower the difficulty level.

But don’t let that fool you into thinking that this problem won’t be challenging. Your challenge is to solve this puzzle in under a minute! You have the following equation as your starting point:

(x²/y²) + (y²/x²) = 287

Given this equation, your challenge is to figure out the right-hand side of the following equation:

[(x/y) + (y/x)]³ = ??

Hint:

If you land on the right idea, this problem is so ridiculously easy that the one-minute time limit is probably overkill.

Spoiler Alert:

If you wish to solve this puzzle on your own, I suggest that you tune off of this essay and do so now. Just remember to start your timer as you begin solving the problem.

If you are unable to solve it in under a minute (or wish to compare your procedure with mine), you can return to this essay and continue reading. Beyond this section, I will be explicitly discussing solutions to this puzzle.

This essay is supported by Generatebg

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The Starting Point to Solve the Tricky Algebra Problem

The first realization that helps us is that when both the numerator and the denominator of a fraction are raised to the same exponent, it is equal to the fraction involving just the bases raised to the common exponent as follows:

x²/y² = (x/y)²

When we apply this realization to the originally given equation, we get the following expression:

(x²/y²) + (y²/x²) = (x/y)² + (y/x)² = 287

The middle of this expression ((x/y)² + (y/x)²) is of the form (a² + b²). We know that the binomial square formula is as follows:

(a + b)² = a² + b² + 2ab

Let us now suppose the following:

x/y = a; y/x = b

When we apply the binomial square formula to our supposition, we end up with the following expression:

This right-hand side of the final expression is interesting for us. So, let us keep this result at the back of our minds for now.

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How to Really Solve This Tricky Algebra Problem

To make use of our previous result, let us see what happens when we add ‘2’ to both the sides of the originally given equation:

Now, recall our result from earlier. The left-hand side of this expression is nothing but [(x/y) + (y/x)]². The right-hand side of this expression is nothing but |+-17|². We can solve this expression further by taking the square root on both the sides as follows:

We are now at the home stretch. All we need to do is raise both sides of the above expression to the third power. When we do so, we end up with two possible solutions to the problem as follows:

There you have it! We have now arrived at the solution to the problem.

If you are a beginner and felt that this puzzle was too easy for you, do let me know in the comments. Your feedback would help me design/scavenge problems/puzzles for the future!

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Further reading that might interest you: Why Is Negative Times Negative Really Positive? and What Really Happens When You Divide By Zero?

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