Answer:
[tex]\displaystyle c = -\frac{35}{4} = -8.75[/tex].
Step-by-step explanation:
Let the smaller root to this equation be [tex]m[/tex]. The larger one will equal [tex]m + 6[/tex].
By the factor theorem, this equation is equivalent to
[tex]a(x - m)(x - (m+6))= 0[/tex], where [tex]a \ne 0[/tex].
Expand this expression:
[tex]a\cdot x^{2} - a(2m + 6)\cdot x + a(m^{2} + 6m) =0[/tex].
This equation and the one in the question shall differ only by the multiple of a non-zero constant. It will be helpful if that constant is equal to [tex]1[/tex]. That way, all constants in the two equations will be equal; [tex](m^{2} + 6m)[/tex] will be equal to [tex]c[/tex].
Compare this equation and the one in the question:
The coefficient of [tex]x^{2}[/tex] in the question is [tex]1[/tex] (which is omitted.) The coefficient of [tex]x^{2}[/tex] in this equation is [tex]a[/tex]. If all corresponding coefficients in the two equations are equal to each other, these two coefficients shall also be equal to each other. Therefore [tex]a = 1[/tex].
This equation will become:
[tex]x^{2} - (2m + 6)\cdot x + (m^{2} + 6m) =0[/tex].
Similarly, for the coefficient of [tex]x[/tex],
[tex]\displaystyle -(2m +6) = 1[/tex].
[tex]\displaystyle m = -\frac{7}{2}[/tex].
This equation will become:
[tex]x^{2} + x + \underbrace{\left(-\frac{35}{4}\right)}_{c} =0[/tex].
[tex]c[/tex] is the value of the constant term of this quadratic equation.
