HW help

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)

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.