Find all real numbers a such that a2=a−2. Explain why this does not violate the exponential function property which states ax=ay if and only if x=y.
Find all real numbers a such that a2=a−2. Explain why this does not violate the exponential function property which states ax=ay if and only if x=y.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Find all real numbers a such that
a2=a−2.
Explain why this does not violate the exponential function property which states
ax=ay
if and only if
x=y.
Expert Solution
Step 1
a2 = a-2 implies, a2 = 1/a2
Then, a4 = 1
So, a4 - 1 = 0
i.e. (a² - 1)(a²+1) = 0
Since, a² + 1 > 0 for all real a, we only have,
a² - 1 = 0
i.e. a = 1, - 1
So, the solution is, a = 1 & - 1
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