binomial square within a square root equals an absolute binomial (number)?

A square root can be either positive or negative.

So, for instance, √9 can be either +3 or -3, and this is commonly written as ±3, but can also be expressed as |3|, which also means it can be either positive or negative.

However, if you see √x without any further indication, it is taken to mean the positive root. It must be made clear if the negative root is intended. In solving some real-life problems which require a quadratic equation, it is clear from the statement of the problem whether the correct answer is the positive or the negative one.

So although (2x + 3) is a perfectly good solution for √(2x+3)² (in fact it is the only one, given that you commence with (2x + 3) inside the root), the complete solution for √(2x+3)² is ±(2x + 3), or |(2x + 3)|.

Square root is always positive.

For example, let x = -2 ---> 2(-2) + 3 = -1

4(-2)^2 + 12(-2) + 9 = 16 - 24 + 9 = 1

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Simplest example of root(x^2) = absvalue(x)

Let x = -3, x^2 = 9, root(x^2) = root(9) = 3 which is the absolute value of -3. Always works this way. When you see root(x^2), think absvalue(x)

the absolute value entails both negative and positive value.

|a| = |-a| = a

the square root of a number is both positive and negative. eg.

(2)^2 = 4

(-2)^2 = 4

so √4 = 4 or -4.

with the absolute value, we only consider the magnitude of the number