The sum of the square of a number and 12 times the number is -27. What is the smaller possible value of this number?
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:
- x = (-12 + 6) / 2 = -6 / 2 = -3
- 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.
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