Difference between revisions of "005 Sample Final A, Question 8"

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(Created page with "''' Question ''' Solve the following equation,      <math> 3^{2x} + 3^x -2 = 0 </math> {| class="mw-collapsible mw-collapsed" style = "text-align:left;"...")
 
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! Final Answers
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! Step 1:
 
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|a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, <math>a_n = (-a)^n</math>
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| Start by rewriting <math>3^{2x} = \left(3^x\right)^2</math> and make the substitution <math>y = 3^x</math>
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! Step 2:
 
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|b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
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| After substitution we get <math>y^2 + y - 2 = (y + 2)(y - 1)</math>
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! Step 3:
 
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|c) False. <math>y = x^2</math> does not have an inverse.
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| Now we have to find the zeros of <math>3^x + 2 = 0</math> and <math>3^x - 1 = 0</math>. We do this by first isolating <math>3^x</math> in both equations.
 
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|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
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|So <math>3^x = -2</math> and <math>3^x = 1</math>
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! Step 4:
 
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|e) True.
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| We observe that <math>3^x = -2</math> has no solutions. We can solve <math>3^x = 1</math> by taking <math>log_3</math> of both sides.
 
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|f) False.
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|This gives<math>\log_3\left(3^x\right) = x = \log_3(1) = 0</math>
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! Final Answer:
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| x = 0
 
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Revision as of 14:28, 17 May 2015

Question Solve the following equation,      Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^{2x} + 3^x -2 = 0 }


Step 1:
Start by rewriting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^{2x} = \left(3^x\right)^2} and make the substitution Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = 3^x}
Step 2:
After substitution we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2 + y - 2 = (y + 2)(y - 1)}
Step 3:
Now we have to find the zeros of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^x + 2 = 0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^x - 1 = 0} . We do this by first isolating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^x} in both equations.
So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^x = -2} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^x = 1}
Step 4:
We observe that has no solutions. We can solve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^x = 1} by taking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle log_3} of both sides.
This givesFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log_3\left(3^x\right) = x = \log_3(1) = 0}
Final Answer:
x = 0
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