Consider the following system of inequalities:

{(c−1)x^2+2cx+c+4≤0
cx^2+2(c+1)x+(c+1)≥0
The sum of all real values of c, such that the system has a unique solution, can be written as ab, where a and b are coprime positive integers. What is the value of a+b?

Details and assumptions
c can be negative.

The system has a unique solution if there is only 1 real value x which is satisfied throughout.

19

thanx

To find the sum of all real values of c such that the system of inequalities has a unique solution, we need to analyze the conditions under which the inequalities are both satisfied.

First, let's consider the first inequality: (c−1)x^2 + 2cx + c + 4 ≤ 0.

To determine when this inequality is true, we need to find the values of x that satisfy it. We can start by finding the discriminant of the quadratic equation: Δ = b^2 - 4ac.

In this case, a = (c−1), b = 2c, and c = (c + 4). Plugging these values into the discriminant formula, we get:

Δ = (2c)^2 - 4(c−1)(c+4)

Simplifying further, we have:

Δ = 4c^2 - 4(c^2 + 3c - 4)
= 4c^2 - 4c^2 - 12c + 16
= -12c + 16

Since the discriminant needs to be greater than or equal to zero for real solutions, we have:

-12c + 16 ≥ 0

Solving this inequality, we find:

-12c ≥ -16
c ≤ 16/12
c ≤ 4/3

Therefore, c must be less than or equal to 4/3 for the first inequality to hold true.

Now let's consider the second inequality: cx^2 + 2(c+1)x + (c+1) ≥ 0.

We can again find the discriminant, using a = c, b = 2(c+1), and c = (c + 1):

Δ = (2(c+1))^2 - 4c(c+1)

Simplifying, we have:

Δ = 4(c^2 + 2c + 1) - 4c(c+1)
= 4c^2 + 8c + 4 - 4c^2 - 4c
= 4c^2 + 4c^2 + 8c - 4c - 4c
= 8c^2

To have real solutions, the discriminant needs to be greater than or equal to zero:

8c^2 ≥ 0

Since the coefficient of c^2 is positive, this inequality holds for all real values of c.

Now, we need to find the values of c that satisfy both inequalities simultaneously:

c ≤ 4/3 (from the first inequality)
(all real values of c) (from the second inequality)

Since the first inequality restricts the values of c, the intersection of the solution sets of both inequalities will be the values that satisfy both conditions.

Given that the set of real numbers includes all possible values of c, the intersection is simply the range for c ≤ 4/3.

Therefore, the sum of all real values of c such that the system has a unique solution is the sum of all values of c less than or equal to 4/3. This sum can be expressed as 4/3.

In conclusion, a = 4 and b = 3. The sum of a and b is 4 + 3 = 7.