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

tommarvoloriddle Jul 1, 2019

#1**+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)

CPhill Jul 1, 2019

#7**+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

#2**+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

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

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

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.

Melody Jul 1, 2019