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sum, reciprocals, squares (Posted on 2022-03-22)

The sum of two numbers is 1.

The sum of the reciprocals of the numbers is 1.

Find the sum of the squares of the numbers (without finding the numbers themselves.)

What do you notice?

No Solution Yet Submitted by Jer

Rating: 4.0000 (1 votes)

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Findings

| Comment 2 of 4 |

x + y = 1

1/x + 1/y = 1

(x+y)*(1/x + 1/y) = 1 + x/y + 1 + y/x

1 = 2 + x/y + y/x

x/y + y/x = -1

x^2 + y^2 = -xy

Verification:

From Wolfram Alpha:

x=1/2 + sqrt(3/4) * i, y=1/2 - sqrt(3/4) * i

x=1/2 - sqrt(3/4) * i, y=1/2 + sqrt(3/4) * i

Multiplying corresponding x and y in MATLAB gives 1.

Adding their squares gives -1, verifying that x^2 + y^2 = -xy.

Posted by Charlie on 2022-03-22 08:58:45

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