Question

simplify
(2+√3)(-2-√3)​

Answer 1

Answer:

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Answer 2

Answer:

What is the solution of (2+√3) (2-√3)?

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The answer is 1.

1. Explanation:

Using the rule: (a+b) (a-b) = a^2 - b^2

Our equation: (2+sqrt3) (2-sqrt3) = 2^2 - sqrt3^2

= 4 - 3 = 1

Notes:

1.1 The definition of square root:

sqrt(n) * sqrt(n) = n.

This is by the definition of square root:

…. If r = sqrt(n)

…. then r X r = n.

1.2 Any n which is a result of squaring - is necessarily a positive number.

1.3 The square root of three

Unimportant information follows:

The ancient Greeks proved that the square root of 3 is an irrational number,

that cannot be obtained by dividing one number by another.

In decimal notation you will always need to “a

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One of the most useful identities is the factorization of a difference of squares: a2−b2=(a+b)(a−b) . (Just multiply it out. FOIL works fine: the outer and inner terms cancel.)

The given problem is the factorization of the difference of squares a2−b2 with a=2 and b=3–√ . Since a2=4 and b2=3 , we get (2+3–√)(2−3–√)=22−(3–√)2=4−3=1 .

The given expression is equal to 1.

Frequently one wants to factorize a sum of squares: a2+b2 . This is not possible using real numbers only. But a2+b2=(a+ib)(a−ib) where i=−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.