Solve without squaring both sides: 5 - (2/x) = √(5 - 2/x).

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Start by isolating the square root term on one side of the equation. Subtract 5 from both sides to get:

-(2/x) = \(\sqrt{5 - 2/x}\) - 5

Multiply through by -1 to simplify the left-hand side:

2/x = 5 - \(\sqrt{5 - 2/x}\)

To eliminate the fraction, multiply through by

x

(assuming

x \(\neq\) 0

2 = x(5 - \(\sqrt{5 - 2/x}\))

Distribute

x

to both terms inside the parentheses:

2 = 5x - x\(\sqrt{5 - 2/x}\)

At this point, you can analyze the equation further to isolate the square root term or simplify, but avoid squaring both sides as per the problem's instructions. Consider alternative methods such as substitution or factoring to proceed.

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주요 개념

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Rational Expressions

Rational expressions are fractions that contain polynomials in the numerator and denominator. Understanding how to manipulate these expressions is crucial for solving equations involving variables in the denominator, such as the term 2/x in the given equation. Simplifying and finding common denominators can help isolate variables and solve for x.

Square Roots

Square roots are mathematical operations that determine a number which, when multiplied by itself, gives the original number. In the context of the equation, recognizing how to work with square roots without squaring both sides is essential. This involves understanding properties of square roots, such as the fact that √a = b implies a = b², but also that both sides must remain valid under the operations performed.

Isolating Variables

Isolating variables is a fundamental algebraic technique used to solve equations. This involves rearranging the equation to get the variable of interest on one side. In the given problem, isolating the term involving x will allow for a clearer path to finding the solution without squaring both sides, which can introduce extraneous solutions.