2021 AMC 10B Problems/Problem 15
The real number satisfies the equation . What is the value of
We square to get . We subtract 2 on both sides for and square again, and see that so . We can divide our original expression of by to get that it is equal to . Therefore because is 7, it is equal to .
Multiplying both sides by and using the quadratic formula, we get . We can assume that it is , and notice that this is also a solution the equation , i.e. we have . Repeatedly using this on the given (you can also just note Fibonacci numbers),
We can immediately note that the exponents of are an arithmetic sequence, so they are symmetric around the middle term. So, . We can see that since , and therefore . Continuing from here, we get , so . We don't even need to find what is! This is since is evidently , which is our answer.
We begin by multiplying by , resulting in . Now we see this equation: . The terms all have in common, so we can factor that out, and what we're looking for becomes . Looking back to our original equation, we have , which is equal to . Using this, we can evaluate to be , and we see that there is another , so we put substitute it in again, resulting in . Using the same way, we find that is . We put this into , resulting in , so the answer is .
The equation we are given is Yuck. Fractions and radicals! We multiply both sides by square, and re-arrange to get Now, let us consider the expression we wish to acquire. Factoring out we have Then, we notice that Furthermore, Thus, our answer is ~peace09
Video Solution by OmegaLearn (Algebraic Manipulations and Symmetric Polynomials)
Video Solution by Interstigation (Simple Silly Bashing)
Video Solution by TheBeautyofMath
Not the most efficient method, but gets the job done.
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