Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.0 AMC 10/12 Find the answer Philippines

Problem:

There are two distinct real numbers which are larger than their reciprocals by $2$ . Find the product of these numbers.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Let $x$ be one of these real numbers. We have $x = \frac{1}{x} + 2$ , which is equivalent to $x^2 - 2x - 1 = 0$ . The required real numbers are the two distinct roots of this quadratic equation, which have a product of $-1$ .

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