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avatar+1681 

 I think the answer is 2. If it is wrong, can you tell me why? And I used guesswork, so is there a more efficient way? And can you help me out by using that more efficient way and telling me how below?

 

Thank you

- tommarvoloriddle

 Jul 1, 2019
edited by tommarvoloriddle  Jul 1, 2019
 #1
avatar+105466 
+2

We can write

 

144

____   =     10   + a       multiply both sides by a

 a

 

 

144  = 10a  + a^2       rearrange as

 

a^2 + 10a  - 144  = 0

 

We need two integers that sum to  10   and multiply to  -144

 

These are  18  and - 8

 

So

 

(a + 18) (a - 8)  = 0

 

Set each factor to 0  and solve for a

 

a + 18  = 0                                     a - 8  = 0

a  = -18   (reject)                             a = 8     (this is the solution)

 

 

 

 

cool cool cool

 Jul 1, 2019
 #4
avatar+1681 
+10

Thank you CPHILL

tommarvoloriddle  Jul 1, 2019
 #5
avatar+105466 
0

Welcome.....

 

 

cool cool cool

CPhill  Jul 1, 2019
 #6
avatar+1681 
+10

:)

-tommarvoloriddle

tommarvoloriddle  Jul 1, 2019
 #7
avatar+2505 
+2

When you thought the answer is 2, this was your mistake:

 

 

You thought \(\Delta\) could be a two digit number, and assumed it was 12.

 

However, in the problem, it states "1\(\Delta\)". It cannot be "112"

 

So in the future, be careful, remember that when "1\(\Delta\)" happens, \(\Delta\) must be SINGLE digit number.

CalculatorUser  Jul 2, 2019
 #8
avatar+1681 
+5

:) whooooops

tommarvoloriddle  Jul 2, 2019
 #2
avatar+106080 
+2

Did you test 2?

144/2=72   there is not a 1 in the units column so it cannot be correct.

You could just try all the possible answers, there are only 9 of them, 1 through 9 but better still, do it as I have below  laugh

 

 

 

\(let \;\;x=\triangle\)

 

\(\frac{144}{x}=10+x\\ 144=x^2+10x\\ x^2+10x-144=0\\ x=\frac{-10\pm\sqrt{100+576}}{2}\\ x=\frac{-10\pm\sqrt{676}}{2}\\ x=\frac{-10\pm26}{2}\\ x=-5\pm13\\ x = -18\;\;or\;\;8\)

 

but x is a 1 digit number so it is 8

 

\(\triangle \;\;is\;\;8\)

 

check:   144/8=18     there is a one in the tens column and an 8 in the units so it works.   laugh

.
 Jul 1, 2019
edited by Melody  Jul 1, 2019
 #3
avatar+1681 
+11

Whoops. Thank you Melody!

tommarvoloriddle  Jul 1, 2019

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