005 Sample Final A, Question 8 - Revision history

Matt Lee at 03:25, 22 May 2015

2015-05-22T03:25:16Z

Matt Lee at 03:24, 22 May 2015

2015-05-22T03:24:51Z

Matt Lee at 03:24, 22 May 2015

2015-05-22T03:24:37Z

Matt Lee at 22:28, 17 May 2015

2015-05-17T22:28:07Z

Matt Lee: Created page with "''' Question ''' Solve the following equation, $3^{2x} + 3^x -2 = 0$ {| class="mw-collapsible mw-collapsed" style = "text-align:left;"..."

2015-05-01T04:24:04Z

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;"..."

New page

''' Question ''' Solve the following equation,      <math> 3^{2x} + 3^x -2 = 0 </math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answers
|-
|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>
|-
|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.
|-
|c) False. <math>y = x^2</math> does not have an inverse.
|-
|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
|-
|e) True.
|-
|f) False.
|}

@@ Line 9: / Line 9: @@
 |Answer:
 |-
-|1) Substitute y = <math>3^x</math> to change the original equation into <math>y^2 + y - 2 = 0</math>
+|1) Substitute <math>y = 3^x</math> to change the original equation into <math>y^2 + y - 2 = 0</math>
 |}

← Older revision		Revision as of 03:24, 22 May 2015
Line 1:		Line 1:
	''' Question ''' Solve the following equation,      <math> 3^{2x} + 3^x -2 = 0 </math>		''' Question ''' Solve the following equation,      <math> 3^{2x} + 3^x -2 = 0 </math>
		+
		+	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"
		+	!Foundations
		+	\|-
		+	\|1) What substitution can we make to simplify the problem?
		+	\|-
		+	\|Answer:
		+	\|-
		+	\|1) Substitute y = <math>3^x</math> to change the original equation into <math>y^2 + y - 2 = 0</math>
		+	\|}

@@ Line 3: / Line 3: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Final Answers
+! Step 1:
 |-
-|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>
+| Start by rewriting <math>3^{2x} = \left(3^x\right)^2</math> and make the substitution <math>y = 3^x</math>
 |-
-|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.
+| After substitution we get <math>y^2 + y - 2 = (y + 2)(y - 1)</math>
 |-
-|c) False. <math>y = x^2</math> does not have an inverse.
+| 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.
 |-
-|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
+|So <math>3^x = -2</math> and <math>3^x = 1</math>
 |-
-|e) True.
+| 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.
 |-
-|f) False.
+|This gives<math>\log_3\left(3^x\right) = x = \log_3(1) = 0</math>
 |}

005 Sample Final A, Question 8 - Revision history

Matt Lee at 03:25, 22 May 2015

Matt Lee at 03:24, 22 May 2015

Matt Lee at 03:24, 22 May 2015

Matt Lee at 22:28, 17 May 2015

Matt Lee: Created page with "''' Question ''' Solve the following equation, 3^{2x} + 3^x -2 = 0 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"..."

Matt Lee: Created page with "''' Question ''' Solve the following equation, $3^{2x} + 3^x -2 = 0$ {| class="mw-collapsible mw-collapsed" style = "text-align:left;"..."