If a is a real number and a cubic equation

f(x)=x^3−12x^2+(a+20)x−2a
has a repeated root, what is the sum of all the possible values of a ?

The equation mx2 + 12x + 9m = 0 has different roots for different values of m. For what values of m will the equation have one repeated root?

To find the sum of all possible values of 'a' for which the given cubic equation has a repeated root, we can make use of the fact that a cubic equation with a repeated root can be expressed in the form:

f(x) = (x - r)^2 * (x - s)

where 'r' and 's' are the repeated roots.

In our case, the cubic equation is given as:
f(x) = x^3 - 12x^2 + (a+20)x - 2a

Expanding the above equation using the formulas for the product of binomials, we get:
f(x) = (x^2 - 2rx + r^2) * (x - s)

Comparing the coefficients of similar powers of 'x', we can find the values of 'r' and 's':

For the x^2 term:
-2r = -12
=> r = 6

For the constant term:
r^2 * s = 2a

Now, we need to find the possible values of 'a' such that r^2 * s = 2a.

Substituting the value of 'r', we have:
(6)^2 * s = 2a
36s = 2a
18s = a

Hence, the sum of all possible values of 'a' is equal to the sum of all possible values of 18s.

To find the sum of all possible values of 18s, we need to consider the possible values of 's'. Since 's' can take any real value, there are infinitely many possible values for 's'. Therefore, there is no finite sum of all possible values of 'a'.

In conclusion, the sum of all possible values of 'a' is not defined or does not have a finite value.