# What are we learning ?

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" text_font_size="20px" text_letter_spacing="1px" text_line_height="1.5em" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Competency in Focus: Discriminant of the quadratic equation and basic inequalities  This problem from I.S.I. B.Stat. Entrance 2017 is a problem on quadratic equation and some basic inequalities .

# Next understand the problem

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" text_font_size="20px" text_letter_spacing="1px" text_line_height="1.5em" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Let a , b , c be real numbers such that a+b+c<0. Suppose that equation $ax^2+bx+c=0$ has imaginary roots .Then ,[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="4.0"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.1" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.1"]I.S.I. B.Stat. Entrance 2016, UGA Problem 26[/et_pb_accordion_item][et_pb_accordion_item title="Key Competency" _builder_version="4.1" open="off"]Discriminant of the quadratic equation and basic inequalities [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.1" open="off"]6/10[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.0.9" open="off"]Challenges and Thrills in Pre College Mathematics Excursion Of Mathematics

[/et_pb_text][et_pb_tabs _builder_version="4.1"][et_pb_tab title="HINT 0" _builder_version="4.1"]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title="HINT 1" _builder_version="4.1"]Given that the quadratic equation $ax^2+bx+c=0$ has imaginary roots then what can we say about it's discriminant ?[/et_pb_tab][et_pb_tab title="HINT 2" _builder_version="4.1"]Discriminant of the quadratic equation having imaginary roots is less than zero . So, we have $D<0\Rightarrow 0\leq b^2 <4ac$
So $a,c$ have same sign.[/et_pb_tab][et_pb_tab title="HINT 3" _builder_version="4.1"]Also,$b^2<(a+c)^2-(a-c)^2$
$\Rightarrow 0\leq (a-c)^2<(a+b+c)(a+c-b)$
$\Rightarrow a+c, as $a+b+c<0$. Now we will check the options.[/et_pb_tab][et_pb_tab title="HINT 4" _builder_version="4.1"]This shows, if $a>0,c>0$ then $b>0$, contradicting $a+b+c<0$. Hence $a<0,c<0$.[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]