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Question from AMTI 2015

Note by Saran Balachandar
2 years, 1 month ago

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  Easy Math Editor

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**bold** or __bold__ bold

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1. numbered
2. list

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  2. list
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[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

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Hint:- \( (a+b)(b+c)(c+a) = (a+b+c)(ab+cb+ca) - abc \)

Siddhartha Srivastava - 2 years, 1 month ago

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Can you solve it??

Saran Balachandar - 2 years, 1 month ago

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Using the hint given by Siddhartha Srivatsava, you can substitute the values of

\((a+b+c)(ab+cb+ca) - abc\) by Vieta's formula

Brilliant Member - 2 years, 1 month ago

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@Siddhartha Srivastava Nice hint. Here is another approach which doesn't require knowing the algebraic identity.

\( (a+b)(a+c)(b+c) = ( s - a)(s-b) ( s-c) = f(s) \) where \( s = a+b+c \) and \( f(x) = x^3 - 7x^2 - 6x + 5 \).

Calvin Lin Staff - 2 years, 1 month ago

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\(If\quad a\quad cubic\quad polynomial\quad P(x)=ax^{ 3 }+bx^{ 2 }+cx+d\quad satisifies\quad the\quad condtition\quad P(X)=0,\quad \\ then\quad the\quad roots\quad x_{ 1 },x_{ 2 },x_{ 3 }\quad of\quad the\quad equation\quad P(x)=0\quad satisfy\quad x_{ 1 }+x_{ 2 }+x_{ 3 }=-\frac { b }{ a } ,\quad \\ x_{ 1 }x_{ 2 }+x_{ 1 }x_{ 3 }+x_{ 2 }x_{ 3 }=\frac { c }{ a } ,\quad x_{ 1 }x_{ 2 }x_{ 3 }=-\frac { d }{ a } .\\ (a+b)(b+c)(c+a)\quad can\quad be\quad written\quad as\quad (a+b+c)(ab+bc+ca)-abc\\ \Rightarrow \quad (7)(-6)-(-5)\quad \Rightarrow \quad -37\)

Saran Balachandar - 2 years, 1 month ago

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What have you tried? Where are you stuck?

Calvin Lin Staff - 2 years, 1 month ago

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Thanks ! Got It.

Saran Balachandar - 2 years, 1 month ago

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{7x (-6)}-(-5)=-42+5= -37

Pranjal Prashant - 2 years, 1 month ago

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