Solve each equation. √(4x+1)-√(x-1)=2

검증된 단계별 안내

Identify the equation given: \(\sqrt{4x+1} - \sqrt{x-1} = 2\). Our goal is to solve for \(x\).

Isolate one of the square root terms to one side of the equation. For example, add \(\sqrt{x-1}\) to both sides to get \(\sqrt{4x+1} = 2 + \sqrt{x-1}\).

Square both sides of the equation to eliminate the square root on the left. This gives: \(\left(\sqrt{4x+1}\right)^2 = \left(2 + \sqrt{x-1}\right)^2\).

Simplify both sides: the left side becomes \(4x + 1\), and the right side expands using the formula \((a+b)^2 = a^2 + 2ab + b^2\) to \(4 + 4\sqrt{x-1} + (x - 1)\).

Rearrange the equation to isolate the remaining square root term, then square both sides again to eliminate it. After that, solve the resulting polynomial equation for \(x\), and check all solutions in the original equation to avoid extraneous roots.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Square Root Functions and Radicals

Square root functions involve expressions under a radical sign (√). Understanding how to manipulate and simplify radicals is essential, especially when isolating terms or combining like radicals in an equation.

Isolating and Squaring Both Sides

To solve equations with square roots, isolate one radical on one side and then square both sides to eliminate the square root. This process may need to be repeated if multiple radicals remain, but it can introduce extraneous solutions.

Checking for Extraneous Solutions

After solving, substitute solutions back into the original equation to verify their validity. Squaring both sides can create extraneous solutions that do not satisfy the original equation, so this step ensures only true solutions are accepted.