91Ó°ÊÓ

Understanding how to simplify square root expressions is crucial for solving various algebraic problems. To simplify a square root expression like \(\sqrt{2+4x-x^2}\), we should try to manipulate the terms under the square root sign into a perfect square.

A perfect square is an expression that can be written as \(a^2\), where \(a\) is a real number. By rearranging \(2+4x-x^2\), we can rewrite it as \(\sqrt{(2-x)^2}\), which simplifies further to \(\left| 2-x \right|\). This simplification is possible because the square root of a squared quantity is the absolute value of that quantity.

When dealing with absolute value during this simplification, we always consider two scenarios: one where \(x\) is less than 2, leading to \( -(2-x)\), and one where \(x\) is greater than or equal to 2, resulting in \(x-2\). Thus, the simplification of square root expressions requires understanding the properties of both perfect squares and absolute values.

The absolute value measures the magnitude of a number or expression, regardless of its sign. It's denoted by vertical bars, like this: \( |x| \). Understanding its properties is fundamental in algebra.

Firstly, the absolute value of a real number is always non-negative. This means \( |x| \geq 0\) for any real number \(x\). Secondly, the absolute value of a product is the product of the absolute values \( |ab| = |a| \cdot |b| \). Lastly, for the equation \( |x| = a \), where \(a \geq 0\), there are two possible solutions for \(x\): \(x = a\) or \(x = -a\).

In the context of simplifying square root expressions, we employ the property that \( \sqrt{x^2} = |x| \). This property is crucial when we want to rewrite expressions under the square root in forms like \( (2-x)^2 \), leading to \( |2-x| \). Understanding the scenarios where the absolute value becomes positive or negative helps us accurately simplify these expressions.

Understanding the domain of the inverse sine function, or \( \sin^{-1} \), is important when solving trigonometric equations. The function \( \sin^{-1}(x) \) (also denoted as arcsin) is defined for all real numbers \(x\) such that \( -1 \leq x \leq 1 \). This is because the sine function, of which \( \sin^{-1} \) is the inverse, has a range of \( [-1, 1] \).

When solving equations that involve \( \sin^{-1} \), it's important to ensure that the argument of the inverse sine is within this interval. If not, the expression does not represent a valid input for the function and thus is outside its domain.

In our exercise, \( \sin^{-1}\frac{x-2}{\sqrt{6}} \) implies that \( \frac{x-2}{\sqrt{6}} \) must lie between -1 and 1 for the expression to be valid. By solving the inequality \( -\sqrt{6}+2 \leq x \leq \sqrt{6}+2 \) we are effectively finding the values of \(x\) that keep the argument of the inverse sine within its domain. It's essential to check the domain constraints when dealing with inverse trigonometric functions to avoid invalid solutions.