Solve each equation. √x-√x+3= -1

Verified step by step guidance

1. First, carefully rewrite the equation to clearly identify each term: \(\sqrt{x} - \sqrt{x + 3} = -1\).
2. Isolate one of the square root terms to one side of the equation. 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. Remember to apply the formula \((a - b)^2 = a^2 - 2ab + b^2\) when squaring the right side: \(\left(\sqrt{x}\right)^2 = \left(\sqrt{x + 3} - 1\right)^2\).
4. Simplify both sides after squaring: the left side becomes \(x\), and the right side expands to \((x + 3) - 2\sqrt{x + 3} + 1\).
5. Isolate the remaining square root term and square both sides again to completely eliminate the radicals. Then solve the resulting polynomial equation for \(x\). Remember to check your solutions in 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 expressions involving square roots is essential, especially recognizing that the square root function outputs only non-negative values.

Solving Radical Equations

Solving equations with radicals often requires isolating the radical expression and then squaring both sides to eliminate the square root. Care must be taken to check for extraneous solutions introduced by this process.

Domain Restrictions

The domain of a radical expression is limited to values that keep the radicand (the expression inside the root) non-negative. Identifying these restrictions helps avoid invalid solutions and ensures the equation is solved within its proper context.