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# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]If  $$\alpha$$ is a root of  $$x^2$$ - x +1 = 0 , then $$\alpha^{2018}$$  + $$\alpha^{-2018}$$  is [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.27" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.27" hover_enabled="0" _i="0" _address="0.1.0.0.0"]Sample Questions (MMA) : 2019[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.0.1" open="off"]Quadratic Roots[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.0.2" open="off"]Medium [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.0.3" open="off"]Abstract Algebra - Dummit and Foote [/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="1" _address="0.1.0.1"]

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.27" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" hover_enabled="0" _i="2" _address="0.1.0.2"][et_pb_tab title="Hint 0" _builder_version="3.22.4" _i="0" _address="0.1.0.2.0"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.2.1"]We have to use algebraic expression and expansion e.g-  $$( x + 1) ^{2}$$ = $$x^2$$ + 2x + 1 ; $$( x +1)^{3}$$ = $$x^{3}$$ + 3$$x^2$$ +3x + 1 Can you think about this kind of case a bit more ?[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.2.2"]Note that $$x^3$$+ 1 = ($$x+1$$)($$x^{2} - x + 1$$) So, $$\alpha$$ is a root of $$x^{2} - x + 1$$ we have   ($$\alpha^3$$ + 1) = ($$\alpha+1$$)($$\alpha^{2}$$ - $$\alpha$$ + 1 ) = ( $$\alpha$$ + 1 ) = 0 Can we think along this line ?[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.2.3"]$$\alpha^{3}$$ = -1   ||ly $$\beta^{3}$$ = -1 [ when $$\beta$$ is another root ($$x^{2} - x + 1$$) ] So, ($$x^{2} - x + 1$$) = ( x - $$\alpha$$ )( x - $$\beta$$) = $$x^{2}$$ - ($$\alpha + \beta$$)x + $$\alpha$$$$\beta$$. Can you get the expression of $$\beta$$ in terms of $$\alpha$$ ? Is the cloud getting clear now ?[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27" hover_enabled="0" _i="4" _address="0.1.0.2.4"]$$\alpha$$ + $$\beta$$ = 1 $$\alpha$$$$\beta$$ =  1  implying   $$\beta$$  =  1/ $$\alpha$$  So, $$\alpha^{2018}$$  + $$\alpha^{-2018}$$  = $$\alpha^{2018}$$  + $$\beta^{2018}$$  Note that : $$\alpha^{2}$$  +  $$\beta^{2}$$ =  $$\alpha + \beta )^{2}$$ -2$$\alpha$$$$\beta$$  = 1 - 2$$\alpha$$$$\beta$$  = 1 -2 = -1   $$\alpha^{2018}$$ +   $$\beta^{2018}$$  =  $$\alpha^{3x672}$$ . $$\alpha^{2}$$ + $$\beta^{3x672}$$ . $$\beta^{2}$$ =  ($$\alpha^{3})^{672}$$.$$\alpha^{2}$$ + ($$\beta^{3})^{672}$$.$$\beta^{2}$$ = ($$-1^{672}$$).$$\alpha^{2}$$ + ($$-1^{672}$$).$$\beta^{2}$$ =  $$\alpha^{2}$$  + $$\beta^{2}$$   = -1 [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="3" _address="0.1.0.3"]

# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.27" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" _i="1" _address="0.0.0.1"]If  $$\alpha$$ is a root of  $$x^2$$ - x +1 = 0 , then $$\alpha^{2018}$$  + $$\alpha^{-2018}$$  is [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25" _i="1" _address="0.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||" _i="0" _address="0.1.0"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.27" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px" _i="0" _address="0.1.0.0"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.27" hover_enabled="0" _i="0" _address="0.1.0.0.0"]Sample Questions (MMA) : 2019[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.0.1" open="off"]Quadratic Roots[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.0.2" open="off"]Medium [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.0.3" open="off"]Abstract Algebra - Dummit and Foote [/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="1" _address="0.1.0.1"]

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.27" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff" hover_enabled="0" _i="2" _address="0.1.0.2"][et_pb_tab title="Hint 0" _builder_version="3.22.4" _i="0" _address="0.1.0.2.0"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.27" hover_enabled="0" _i="1" _address="0.1.0.2.1"]We have to use algebraic expression and expansion e.g-  $$( x + 1) ^{2}$$ = $$x^2$$ + 2x + 1 ; $$( x +1)^{3}$$ = $$x^{3}$$ + 3$$x^2$$ +3x + 1 Can you think about this kind of case a bit more ?[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.27" hover_enabled="0" _i="2" _address="0.1.0.2.2"]Note that $$x^3$$+ 1 = ($$x+1$$)($$x^{2} - x + 1$$) So, $$\alpha$$ is a root of $$x^{2} - x + 1$$ we have   ($$\alpha^3$$ + 1) = ($$\alpha+1$$)($$\alpha^{2}$$ - $$\alpha$$ + 1 ) = ( $$\alpha$$ + 1 ) = 0 Can we think along this line ?[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.27" hover_enabled="0" _i="3" _address="0.1.0.2.3"]$$\alpha^{3}$$ = -1   ||ly $$\beta^{3}$$ = -1 [ when $$\beta$$ is another root ($$x^{2} - x + 1$$) ] So, ($$x^{2} - x + 1$$) = ( x - $$\alpha$$ )( x - $$\beta$$) = $$x^{2}$$ - ($$\alpha + \beta$$)x + $$\alpha$$$$\beta$$. Can you get the expression of $$\beta$$ in terms of $$\alpha$$ ? Is the cloud getting clear now ?[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.27" hover_enabled="0" _i="4" _address="0.1.0.2.4"]$$\alpha$$ + $$\beta$$ = 1 $$\alpha$$$$\beta$$ =  1  implying   $$\beta$$  =  1/ $$\alpha$$  So, $$\alpha^{2018}$$  + $$\alpha^{-2018}$$  = $$\alpha^{2018}$$  + $$\beta^{2018}$$  Note that : $$\alpha^{2}$$  +  $$\beta^{2}$$ =  $$\alpha + \beta )^{2}$$ -2$$\alpha$$$$\beta$$  = 1 - 2$$\alpha$$$$\beta$$  = 1 -2 = -1   $$\alpha^{2018}$$ +   $$\beta^{2018}$$  =  $$\alpha^{3x672}$$ . $$\alpha^{2}$$ + $$\beta^{3x672}$$ . $$\beta^{2}$$ =  ($$\alpha^{3})^{672}$$.$$\alpha^{2}$$ + ($$\beta^{3})^{672}$$.$$\beta^{2}$$ = ($$-1^{672}$$).$$\alpha^{2}$$ + ($$-1^{672}$$).$$\beta^{2}$$ =  $$\alpha^{2}$$  + $$\beta^{2}$$   = -1 [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" _i="3" _address="0.1.0.3"]

# Similar Problems

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### 2 comments on “Problems on quadratic roots: ISI MMA 2018 Question 9”

1. Aritra Dey says:

it can be solved more easily.

1. Dey Sarkar Arnab says:

Hi,

Yes, you are right but this method came into my mind first so I typed it down :). You can write your solution so that the readers can see that as well. Thanks a lot for seeing this. Keep reading and don't forget to comment. Your valuable comments/appreciations inspire us to create more contents.

Regards,
Arnab