Solve each equation. √x-√x+3= -1

Verified step by step guidance

1. Rewrite the equation clearly: \(\sqrt{x} - \sqrt{x + 3} = -1\).
2. Isolate one of the square root terms to one side. For example, add \(\sqrt{x + 3}\) to both sides to get \(\sqrt{x} = \sqrt{x + 3} - 1\).
3. Square both sides of the equation to eliminate the square roots. This means squaring \(\sqrt{x}\) and \(\sqrt{x + 3} - 1\) separately: \(\left(\sqrt{x}\right)^2 = \left(\sqrt{x + 3} - 1\right)^2\).
4. Simplify both sides after squaring: the left side becomes \(x\), and the right side expands using the formula \((a - b)^2 = a^2 - 2ab + b^2\), where \(a = \sqrt{x + 3}\) and \(b = 1\).
5. After simplification, solve the resulting equation for \(x\). Remember to check your solutions by substituting back into the original equation because squaring can introduce extraneous solutions.

Key Concepts

Square Roots and Radicals

Square roots represent a value that, when multiplied by itself, gives the original number. Understanding how to manipulate and simplify square roots is essential for solving equations involving radicals.

Isolating Radical Expressions

To solve equations with square roots, isolate the radical on one side before squaring both sides. This step helps eliminate the square root and simplifies the equation into a solvable form.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce solutions that don't satisfy the original equation. Always substitute solutions back into the original equation to verify their validity.