72nd Problem 2016

Algebra Level 3

Find the sum of all the roots of the following equation:

x 2 − 222 x + 2500 = − 20 \LARGE { x }^{ 2 }-222x+2500=-20

Write the sum in the form of a simplified fraction a b \frac { a }{ b }

[If the sum is 2, write it as 2 1 \frac { 2 }{ 1 } ]

[If the sum is 4 2 \frac { 4 }{ 2 } , write it as 2 1 \frac { 2 }{ 1 } ]

Write your answer as |a+b|.

Check out the set: 2016 Problems


The answer is 223.

Angela Fajardo by Angela Fajardo , 19, Philippines

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2 solutions

. .
Feb 25, 2021

x 2 − 222 x + 2500 = − 20 x ^ { 2 } - 222x + 2500 = -20 . x 2 − 222 x + 2520 = 0 x ^ { 2 } - 222x + 2520 = 0 . ( x − 12 ) ( x − 210 ) = 0 ( x - 12 ) ( x - 210 ) = 0 . x = 12 , x = 210 x = 12, x = 210 . 12 + 210 = 222 12 + 210 = 222 . 222 = 222 1 222 = \frac { 222 } { 1 } . 222 + 1 = 223 222 + 1 = \boxed { 223 } .

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Angela Fajardo
Mar 23, 2016

x 2 − 222 x + 2500 = − 20 F o r m u l a f o r S u m o f R o o t s : − b a b = − 222 a = 1 − ( − 222 ) 1 = 222 1 \LARGE { x }^{ 2 }-222x+2500=-20\\ \LARGE Formula\quad for\quad Sum\quad of\quad Roots:\frac { -b }{ a } \\ \LARGE b=-222\\ \LARGE a=1\\ \\ \LARGE \frac { -(-222) }{ 1 } =\frac { 222 }{ 1 }

222 + 1 = 223 222+1=223

222 + 1 = 223 (According to Instructions)

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