find all real solutions:

(2x - 16 = \sqrt{-77 + 9x})

(x =) (if no solution exists, enter dne; if more than one solution exists, enter the solutions separated by commas)

Question

find all real solutions:

(2x - 16 = \sqrt{-77 + 9x})

(x =)   (if no solution exists, enter dne; if more than one solution exists, enter the solutions separated by commas)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Radical Equations", "Solving Quadratic Equations", "Extraneous Solutions" ], "new_concepts": [], "current_concepts": [ "Radical Equations", "Solving Quadratic Equations", "Extraneous Solutions" ] } ## Isolate the radical and square both sides $$ \begin{aligned} 2x - 16 &= \sqrt{-77 + 9x} \ (2x - 16)^2 &= -77 + 9x \ 4x^2 - 64x + 256 &= -77 + 9x \end{aligned} $$ ## Rearrange into standard quadratic form $$ \begin{aligned} 4x^2 - 73x + 333 = 0 \end{aligned} $$ ## Solve the quadratic equation $$ \begin{aligned} x &= \frac{-(-73) \pm \sqrt{(-73)^2 - 4(4)(333)}}{2(4)} \ x &= \frac{73 \pm \sqrt{5329 - 5328}}{8} \ x &= \frac{73 \pm 1}{8} \ x_1 &= \frac{74}{8} = \frac{37}{4} = 9.25 \ x_2 &= \frac{72}{8} = 9 \end{aligned} $$ ## Check for extraneous solutions $$ \begin{aligned} &\text{For } x = 9: \ &\text{LHS} = 2(9) - 16 = 2 \ &\text{RHS} = \sqrt{-77 + 9(9)} = \sqrt{4} = 2 \quad (\text{Valid}) \ \ &\text{For } x = 9.25: \ &\text{LHS} = 2(9.25) - 16 = 2.5 \ &\text{RHS} = \sqrt{-77 + 9(9.25)} = \sqrt{6.25} = 2.5 \quad (\text{Valid}) \end{aligned} $$ Find all real solutions: $2x - 16 = \sqrt{-77 + 9x}$ $x =$ $9, \frac{37}{4}$ (If no solution exists, enter DNE; if more than one solution exists, enter the solutions separated by commas) { "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 and square both sides

Rearrange into standard quadratic form

Solve the quadratic equation

Check for extraneous solutions

Answer: