find the solutions to the equation $10^{2x} + 11 = (x + 6)^2 - 2$. which values are approximate solutions to the equation? select two answers.

-9.6
-7.4
-4.6
-2.4
0.6

Question

find the solutions to the equation $10^{2x} + 11 = (x + 6)^2 - 2$. which values are approximate solutions to the equation? select two answers.

-9.6
-7.4
-4.6
-2.4
0.6

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Solving Exponential Equations Algebraically", "Solving Exponential Equations Graphically" ], "new_concepts": [ "Transcendental Equation Solving" ], "current_concepts": [ "Solving Exponential Equations Algebraically", "Solving Exponential Equations Graphically", "Transcendental Equation Solving" ] } ## Analyze the given equation Using the *Solving Exponential Equations Algebraically* knowledge point $$ 10^{2x} + 11 = (x + 6)^2 - 2 $$ $$ 10^{2x} + 13 = (x + 6)^2 $$ ## Test negative values for exponential term Using the *Solving Exponential Equations Graphically* knowledge point $$ \begin{aligned} &\text{If } x \ll 0, \quad 10^{2x} \approx 0\ &0 + 13 \approx (x + 6)^2\ &(x + 6)^2 \approx 13\ &x + 6 \approx \pm\sqrt{13} \approx \pm 3.61\ &x_1 \approx -6 + 3.61 = -2.39\ &x_2 \approx -6 - 3.61 = -9.61 \end{aligned} $$ ## Evaluate the given options Using the *Solving Exponential Equations* knowledge point $$ \begin{aligned} &\text{For } x = -9.6: \quad 10^{-19.2} + 13 \approx 13, \quad (-9.6 + 6)^2 = (-3.6)^2 = 12.96 \approx 13\ &\text{For } x = -7.4: \quad 10^{-14.8} + 13 \approx 13, \quad (-7.4 + 6)^2 = (-1.4)^2 = 1.96 e 13\ &\text{For } x = -4.6: \quad 10^{-9.2} + 13 \approx 13, \quad (-4.6 + 6)^2 = (1.4)^2 = 1.96 e 13\ &\text{For } x = -2.4: \quad 10^{-4.8} + 13 \approx 13.000016, \quad (-2.4 + 6)^2 = (3.6)^2 = 12.96 \approx 13\ &\text{For } x = 0.6: \quad 10^{1.2} + 13 \approx 15.85 + 13 = 28.85, \quad (0.6 + 6)^2 = 43.56 e 28.85 \end{aligned} $$ ## Select the two approximate solutions Using the *Solving Exponential Equations Graphically* knowledge point $$ x \approx -9.6 \quad \text{and} \quad x \approx -2.4 $$

[ ] -9.6

[ ] -7.4

[ ] -4.6

[ ] -2.4

[ ] 0.6

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QUESTION IMAGE

Question

Explanation:

Analyze the given equation

Test negative values for exponential term

Evaluate the given options

Select the two approximate solutions

Answer: