The sum of the square of a number and 12 times the number is -27. What is the smaller possible value of this number?

প্রশ্নে বলা হচ্ছে, একটি সংখ্যার বর্গ এবং ঐ সংখ্যার 12 গুণের সমষ্টি 27 হলে সংখ্যাটির সম্ভাব্য ক্ষুদ্রতম মান কত?

ধরি, সংখ্যাটি x

প্রশ্নমতে, $x^2+12x=-27$

$\Rightarrow x^2+12x+27=0$

$\Rightarrow x^2+9x+3x+27=0$

$\Rightarrow x(x + 9) + 3(x + 9) = 0$

∴ (x + 9)(x + 3) = 0

এখানে, হয় (x + 9) = 0

∴ x = - 9

অথবা, (x + 3) = 0

∴ x = - 3

এই দুই সংখ্যার মধ্যে ক্ষুদ্রতম হলো- 9.

Let's call the number we're trying to find "x." According to the problem, we have the equation:

x^2 + 12x = -27

Now, let's solve for x. First, move all terms to one side of the equation:

x^2 + 12x + 27 = 0

Now, this is a quadratic equation. We can try to factor it or use the quadratic formula. In this case, factoring might not be straightforward, so we'll use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In our equation, a = 1, b = 12, and c = 27. Plugging these values into the formula:

x = (-12 ± √(12² - 4 * 1 * 27)) / (2 * 1)

x = (-12 ± √(144 - 108)) / 2

x = (-12 ± √36) / 2

Now, simplify further:

x = (-12 ± 6) / 2

Now, we have two possible solutions:

1. x = (-12 + 6) / 2 = -6 / 2 = -3
2. x = (-12 - 6) / 2 = -18 / 2 = -9

So, there are two possible values for x: -3 and -9. Since the problem asks for the smaller possible value, the answer is -9.