solve the equation given. if there are two solutions, enter them separated by a comma. if there is no solution, enter dne (short for does not exist).

$\sqrt{2x + 10} - \sqrt{x + 4} = 1$

$x =$

Question

solve the equation given. if there are two solutions, enter them separated by a comma. if there is no solution, enter dne (short for does not exist).

$\sqrt{2x + 10} - \sqrt{x + 4} = 1$

$x =$

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", "Solving Quadratic Equations" ], "new_concepts": [], "current_concepts": [ "Radical Equations", "Extraneous Solutions", "Solving Quadratic Equations" ] } ## Isolate one radical term Using the **Radical Equations** knowledge point $$ \sqrt{2x + 10} = \sqrt{x + 4} + 1 $$ ## Square both sides of the equation Using the **Radical Equations** knowledge point $$ \begin{aligned} (\sqrt{2x + 10})^2 &= (\sqrt{x + 4} + 1)^2 \ 2x + 10 &= (x + 4) + 2\sqrt{x + 4} + 1 \ 2x + 10 &= x + 5 + 2\sqrt{x + 4} \end{aligned} $$ ## Isolate the remaining radical term Using the **Radical Equations** knowledge point $$ x + 5 = 2\sqrt{x + 4} $$ ## Square both sides again and solve the quadratic equation Using the **Solving Quadratic Equations** knowledge point $$ \begin{aligned} (x + 5)^2 &= (2\sqrt{x + 4})^2 \ x^2 + 10x + 25 &= 4(x + 4) \ x^2 + 10x + 25 &= 4x + 16 \ x^2 + 6x + 9 &= 0 \ (x + 3)^2 &= 0 \ x &= -3 \end{aligned} $$ ## Check for extraneous solutions Using the **Extraneous Solutions** knowledge point $$ \begin{aligned} \text{Substitute } x = -3 \text{ into the original equation:} \ \sqrt{2(-3) + 10} - \sqrt{-3 + 4} &= \sqrt{4} - \sqrt{1} \ &= 2 - 1 \ &= 1 \end{aligned} $$ Since $1 = 1$, the solution $x = -3$ is valid. Solve the equation given. If there are two solutions, enter them separated by a comma. If there is no solution, enter DNE (short for does not exist). $\sqrt{2x + 10} - \sqrt{x + 4} = 1$ $x =$ -3 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Radical Equations" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Isolate one radical term

Square both sides of the equation

Isolate the remaining radical term

Square both sides again and solve the quadratic equation

Check for extraneous solutions

Answer: