(√2/5+√3/2)(√2+√3) simplify by using identity
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Answer:
What is the solution of (2+√3) (2-√3)?
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100+ Answers

Murray Wolinsky
Answered July 5, 2020
One of the most useful identities is the factorization of a difference of squares: a^2 - b^2 = (a + b) (a - b). (Just multiply it out. FOIL works fine: the outer and inner terms vanish.)
The given problem is the factorization of the difference of squares a^2 - b^2 with a = 2 and b = sqrt (3). Since a^2 = 4 and b^2 = 3, we get (2 + sqrt (3)) (2 - sqrt (3)) = 2^2 - (sqrt (3))^2 = 4 - 3 = 1.
The given expression is equal to 1.
Frequently one wants to factorize a sum of squares: a^2 + b^2. This is not possible using real numbers only. But a^2 + b^2 = (a + ib) (a - ib) where i = sqrt (-1), so one can factorize a sum of squares if you use complex numbers. This is a simple example of the fundamental theorem of algebra: take any polynomial (except a constant) in a single variable with real (or even complex) coefficients. That polynomial will have at least one complex root (solution). That statement is not true if one is limited to real numbers. It says that complex numbers are “complete” in a particular way. Admittedly, this matter is a digression from your question, but it’s interesting and may help instill some curiosity about the remarkable complex numbers and their properties.
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