$\text { Prove that all the solutions to the equation } x ^ { 2 } = x + 1 \text { are irrational. }$

Question

Examlex · Accepted Answer

The Answer is This equation is equivalent to (and therefore has the same solutions as)

x ^ { 2 } - x - 1 = 0

. By the quadratic formula, the solutions are exactly

( 1 \pm \sqrt { 5 } ) / 2

. If either of these were a rational number

r

, then we would have

\sqrt { 5 } = \pm ( 2 r - 1 )

. Since the rational numbers are closed under the arithmetic operations, this would tell us that

\sqrt { 5 }

was rational, which we know from this chapter it is not.

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