Quadratic fun-2

As Abhineet has rightly said, to have integral roots, discriminant must be a 0 or a perfect square which can easily be derived by the quadratic formula.

It is only required to find the sum but had the question been finding the values of a too, then Abhineet's sol will be clearly misleading.

${(12+a)}^{2} - 4(12a + 2) = {z}^{2}$ for any integer z.

CASE 1 - When ${(12+a)}^2 - 4(12a) = {12-a}^{2}$

${(12-a)}^{2} - 8 = {z}^{2}$

${(12-a)}^{2} - {z}^{2} = 8$

$(12- a - z)(12- a + z) = 8$

Now 8 = 4 2 or 8 = 2 4 are same in finding solutions for a because its sign(negative or positive) doesn't change. Also, 8 = 8*1 can also not be the case because if it is so, then a will come as decimals.

Therefore,

$12-a -z = 4$ - > $a + z = 8$

$12-a + z = 2$ - > $a - z= 10$

Therefore, a = 9

CASE 2 - When ${(12+a)}^2 - 4(12a) = {a-12}^{2}$

Similarly,

$(a-12-z)(a-12+z) = 8$

$a - 12 - z = 4$ -> $a - z = 16$

$a - 12 + z = 2$ -> $a + z = 14$

Therefore, a = 15

So, the sum of all possible positive integers a is 15 + 9 = $\boxed{24}$

$For\quad a\quad quadratic\quad equation\quad to\quad have\quad integral\quad roots,\quad the\\ discriminant\quad of\quad that\quad equation\quad must\quad be\quad a\quad zero\quad or\quad \\ perfect\quad square.\quad As\quad a\quad perfect\quad square\quad is\quad quite\quad hard\\ to\quad predict,\quad we\quad would\quad take\quad into\quad consideration\quad only\\ D=0,\quad where\quad D\quad is\quad the\quad discriminant\quad of\quad that\quad particular\\ equation.\\ \\ For\quad { (x-a) }(x-12)+2=0\quad to\quad have\quad integral\quad roots:\\ \\ { x }^{ 2 }+x(-12-a)+12a+2=0\\ Discrimnant=0\\ { (-12-a) }^{ 2 }-4(12a+2)=0\\ 144+{ a }^{ 2 }+24a-48a-8=0\\ { a }^{ 2 }-24a+136=0\\ \\ Clearly,\quad the\quad sum\quad of\quad the\quad values\quad of\quad a\quad can\quad be\quad given\quad by\\ Vieta's\quad formula\quad i.e.\quad sum\quad of\quad roots\quad =\quad -\frac { coefficient\quad of\quad x }{ coefficient\quad of\quad { x }^{ 2 } } \\ So,\quad sum\quad of\quad values\quad of\quad a=-\frac { coefficient\quad of\quad { a } }{ coefficient\quad of\quad { a }^{ 2 } } \\ \qquad \qquad \qquad \qquad \qquad \qquad =-\frac { (-24) }{ 1 } =24$