Answer:
[tex]Sum = -81[/tex]
Step-by-step explanation:
See the comment for complete question.
Given
[tex]c = 36[/tex] ----- Constant
No coefficient of x^2
Required:
Determine the sum of all distinct positive integers of the coefficient of x
Reading through the complete question, we can see that the question has 3 terms which are:
x^2 ---- with no coefficient
x ---- with an unknown coefficient
36 ---- constant
So, the equation can be represented as:
[tex]x^2 + ax + 36 = 0[/tex]
Where a is the unknown coefficient
From the question, we understand that the equation has two negative integer solution. This can be represented as:
[tex]x = -\alpha[/tex] and [tex]x = -\beta[/tex]
Using the above roots, the equation can be represented as:
[tex](x + \alpha)(x + \beta) = 0[/tex]
Open brackets
[tex]x^2 + (\alpha + \beta)x + \alpha \beta = 0[/tex]
To compare the above equation to [tex]x^2 + ax + 36 = 0[/tex], we have:
[tex]a = \alpha + \beta[/tex]
[tex]\alpha \beta = 36[/tex]
Where: [tex]\alpha, \beta <0[/tex] and [tex]\alpha \ne \beta[/tex]
The values of [tex]\alpha[/tex] and [tex]\beta[/tex] that satisfy [tex]\alpha \beta = 36[/tex] are:
[tex]\alpha = -1[/tex] and [tex]\beta = -36[/tex]
[tex]\alpha = -2[/tex] and [tex]\beta = -18[/tex]
[tex]\alpha = -3[/tex] and [tex]\beta = -12[/tex]
[tex]\alpha = -4[/tex] and [tex]\beta = -9[/tex]
So, the possible values of a are:
[tex]a = \alpha + \beta[/tex]
When [tex]\alpha = -1[/tex] and [tex]\beta = -36[/tex]
[tex]a = -1 - 36 = -37[/tex]
When [tex]\alpha = -2[/tex] and [tex]\beta = -18[/tex]
[tex]a = -2 - 18 = -20[/tex]
When [tex]\alpha = -3[/tex] and [tex]\beta = -12[/tex]
[tex]a = -3 - 12 = -15[/tex]
When [tex]\alpha = -4[/tex] and [tex]\beta = -9[/tex]
[tex]a = -4 - 9 = -13[/tex]
At this point, we have established that the possible values of a are: -37, -20, -15 and -9.
The required sum is:
[tex]Sum = -37 -20 -15 - 9[/tex]
[tex]Sum = -81[/tex]
