Solve each equation. √x-√x+3= -1

Verified step by step guidance

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

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, especially when isolating variables or simplifying expressions. For example, √x represents the principal square root of x, which is only defined for non-negative values of x.

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, allowing for easier manipulation and solution. In the given equation, isolating the square root terms will help in simplifying the equation and ultimately finding the value of x.

Squaring Both Sides

Squaring both sides of an equation is a common method used to eliminate square roots. This technique involves raising both sides of the equation to the power of two, which can simplify the equation but may introduce extraneous solutions. In the context of the provided equation, applying this method will help in transforming the equation into a polynomial form that can be solved more easily.