Solve each equation. See Examples 4–6. √(x+5) + 2 = √(x-1)

Verified step by step guidance

1. Isolate one of the square root terms by subtracting 2 from both sides: $\sqrt{x+5}=\sqrt{x-1}-2$.
2. Square both sides of the equation to eliminate the square roots: $(\sqrt{x+5}{)}^2=(\sqrt{x-1}-2{)}^2$.
3. Simplify both sides: $x+5=(\sqrt{x-1}{)}^2-4\sqrt{x-1}+4$.
4. Further simplify: $x+5=x-1-4\sqrt{x-1}+4$.
5. Combine like terms and solve for $x$ by isolating the square root term: $6=-4\sqrt{x-1}$.

Key Concepts

Square Roots

Square roots are mathematical operations that find a number which, when multiplied by itself, gives the original number. In the context of the equation, understanding how to manipulate square roots is essential for isolating variables and solving the equation effectively.

Isolating Variables

Isolating variables involves rearranging an equation to get the variable of interest on one side. This is a fundamental technique in algebra that allows for easier solving of equations, particularly when dealing with square roots and other operations.

Squaring Both Sides

Squaring both sides of an equation is a method used to eliminate square roots. By squaring both sides, you can transform the equation into a polynomial form, which is often easier to solve. However, it is important to check for extraneous solutions after this step.