solve the given equation.

$\sqrt{x + 3} + 1 = -x$

$x =$

Question

solve the given equation.

$\sqrt{x + 3} + 1 = -x$

$x =$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Radical Equations", "Extraneous Solutions", "Quadratic Formula" ], "new_concepts": [], "current_concepts": [ "Radical Equations", "Extraneous Solutions", "Quadratic Formula" ] } ## Isolate the radical term Using the *Radical Equations* knowledge point $$ \begin{aligned} \sqrt{x+3} + 1 &= -x \ \sqrt{x+3} &= -x - 1 \end{aligned} $$ ## Square both sides of the equation Using the *Radical Equations* knowledge point $$ \begin{aligned} (\sqrt{x+3})^2 &= (-x - 1)^2 \ x + 3 &= x^2 + 2x + 1 \end{aligned} $$ ## Write in standard quadratic form Using the *Radical Equations* knowledge point $$ \begin{aligned} x^2 + 2x - x + 1 - 3 &= 0 \ x^2 + x - 2 &= 0 \end{aligned} $$ ## Solve the quadratic equation Using the *Quadratic Formula* knowledge point $$ \begin{aligned} (x + 2)(x - 1) &= 0 \ x = -2 \quad &\text{or} \quad x = 1 \end{aligned} $$ ## Check for extraneous solutions Using the *Extraneous Solutions* knowledge point $$ \begin{aligned} &\text{For } x = -2:\ &\sqrt{-2+3} + 1 = \sqrt{1} + 1 = 2 \ &-(-2) = 2 \ &2 = 2 \quad (\text{Valid}) \ \ &\text{For } x = 1:\ &\sqrt{1+3} + 1 = \sqrt{4} + 1 = 3 \ &-(1) = -1 \ &3 eq -1 \quad (\text{Extraneous}) \end{aligned} $$ Solve the given equation. $\sqrt{x + 3} + 1 = -x$ $x =$ $-2$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Radical Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Isolate the radical term

Square both sides of the equation

Write in standard quadratic form

Solve the quadratic equation

Check for extraneous solutions

Answer: