# Question from AMTI 2015

Note by Saran Balachandar
2 years, 8 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

• bulleted
• list

1. numbered
2. list

1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

> 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} $$\boxed{123}$$

Sort by:

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

- 2 years, 8 months ago

Can you solve it??

- 2 years, 8 months ago

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

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

- 2 years, 8 months ago

@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 $$f(x) = x^3 - 7x^2 - 6x + 5$$.

Staff - 2 years, 8 months ago

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

- 2 years, 8 months ago

What have you tried? Where are you stuck?

Staff - 2 years, 8 months ago

Thanks ! Got It.

- 2 years, 8 months ago

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

- 2 years, 8 months ago