Love the Way You Lie

Algebra Level 2

L 2 + O 2 = 122 L + O = 12 V 2 + E 2 = 170 V + E = 14 \begin{aligned} L^2 &+& O^2 = 122 \\ L &+& O = 12 \\ V^2 &+& E^2 = 170 \\ V &+& E = 14 \\ \end{aligned}

What is L O V E LOVE ?

Note that we want to calculate L × O × V × E L \times O \times V \times E .


The answer is 143.

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John Aries Sarza by John Aries Sarza , 26, Philippines

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7 solutions

Chew-Seong Cheong
May 28, 2015

L + O = 12 ⇒ ( L + O ) 2 = 144 ⇒ L 2 + 2 L O + O 2 = 144 ⇒ 2 L O = 144 − ( L 2 + O 2 ) = 144 − 122 = 22 ⇒ L O = 11 L+ O =12 \quad \Rightarrow (L+O)^2 = 144 \quad \Rightarrow L^2+2LO+O^2 = 144 \\ \Rightarrow 2LO = 144 - (L^2+O^2) = 144 - 122 = 22 \quad \Rightarrow LO = 11

Similarly, V E = 1 4 2 − 170 2 = 196 − 170 2 = 26 2 = 13 VE = \dfrac {14^2-170}{2} = \dfrac {196-170}{2} = \dfrac {26}{2} = 13

Therefore, L O V E = 11 × 13 = 143 LOVE = 11\times 13 = \boxed{143}

44 Helpful 1 Interesting 2 Brilliant 0 Confused

143: "I love you". The number of letters in each word corresponds to a digit in the answer. What a wonderful problem!

Linus Setiabrata - 6 years ago

It's a wonderful problem. At first I could not think of those number, the additives of 12 and 14 that when squared and added will result to 122 and 170, thanks that I am familiar to the square of 13 which is 169. It is only then that I realized that the other number is negative which is 1. Another, I'm amazed at the result, 143. Great!

Hubert Hernandez Jr. - 6 years ago

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Thanks for answering...

John Aries Sarza - 5 years, 12 months ago

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Keep posting sir!

Hubert Hernandez Jr. - 5 years, 12 months ago

This is Great! I like your way! But the way I did it: The only 2 number that have square addition of 122 and have product of 121 are 1 and 11 and the same for the second one: square of 1 plus square of 13 has product of 169 and square of 1 plus square of 13 = 170 then 1 × 11 × 1 × 13 = 143 = L O V E 1\times 11\times 1\times 13=143=LOVE

Ghola Kata - 6 years ago

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That is assuming L L , O O , V V and E E are integers

Julian Poon - 6 years ago

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Well Love is natural, isn't it.=?

Ashwin Padaki - 6 years ago

Congrats I was not able to solve. You're explanation was great and simple

Otávio Silveira - 5 years, 8 months ago

What is LOVE? Ans: Go search on google

Mohammad Farhat - 2 years, 8 months ago
John Church
May 31, 2015

Too much thinking, 122 = 121 + 1 and 170 = 169 + 1 so L = 11, O = 1, V = 13 and E=1 as one solution but when you multiply them together you only get 143, the old military WWI secret code to tell someone you love them 1 = I,4=LOVE,3=You.

6 Helpful 0 Interesting 1 Brilliant 0 Confused

Ohhh that's true I make lot of procedure and got the same hahaha. That is thinking out of the box

Antonio Vergara Gomez - 5 years, 8 months ago
Thales Ives
May 31, 2015

O = 12 - L => L^2 + (12 - L)^2 = 122 => L^2 + 144 - 24L + L^2 = 122 => 2L^2 -24L + 22=0 => L^2 - 12L +11 =0

V = 14 - E => (14 - E)^2 + E^2 = 170 => 196 - 28E +E^2 + E^2 = 170 => 2E^2 - 28E + 26 = 0 => E^2 - 14E + 13 = 0

"O" depends on the value of L, if one of them is 1, the other is 11. And the same happens with "V" and "E"

L = {1; 11} O ={1; 11} V = {1; 13} E = {1;13}

L.O.V.E = 143

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

You skipped the most crucial part for solving this problem.

"O" depends on the value of L, if one of them is 1, the other is 11. And the same happens with "V" and "E"

Why is it so?

You should be explain that because L 2 − 12 L + 11 = 0 L^2 - 12 L + 11 = 0 then factorizing it gives ( L − 1 ) ( L − 11 ) = 0 (L-1)(L-11) = 0 . Thus L = 1 , 11 L=1,11 . When L = 11 L=11 , we get O = 12 − 11 = 1 O = 12 - 11 = 1 . Similarly, when L = 1 L = 1 , we get O = 12 − 1 = 11 O = 12 - 1 = 11 .

Can you elaborate for V V and E E too?

Mary Arans
Jun 17, 2018

L = 12 - O. (12-O)^2 + O^2 = 122. 144 - 24O + 2O^2 = 122. -12O + O^2 = -11. O = 11. L = 1. V = 14 - E. (14-E)^2 + E^2 = 170. 196 - 28E + 2E^2 = 170. -14E + E^2 = -13. E = 13. V = 1. LOVE = 1 × 11 × 1 × 13 = 143.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

L^2 + O^2=(L+O)^2-2LO

so, 122=12^2-2LO

122-144=-2LO

-22=-2LO

LO=11

similarly,

V^2+E^2=(V+E)^2-2VE

170=14^2-2VE

170-196=-2VE

-26=-2VE

VE=13

LO*VE=11X13=143

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Ubaidullah Khan
May 31, 2015

L = 11, O=1, V= 13, and E=1.
L+O = 12 and V+E = 14
Square of L+ Square of O = 12 and Square of V+ Square of E = 14
Hence,
L OVE = (1)(13)(1)(11) = 143



0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

But how do you know that "L = 11, O=1, V= 13, and E=1." in the first place? That should be part of your working.

Furthermore, this is not the only solution that satisfies the condition. See Chew-Seong Cheong's solution for a proper approach.

L^2+O^2 = 122 L+O = 12 (L+O)^2 = 144 L^2 + O^2 +2LO = 144 122 + 2LO = 144 LO = 22/2 = 11

V^2 + E^2 = 170 2VE = 196-170 = 26 VE = 13

LO * VE = L O V*E = 11 *13 = 143

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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