A and B are real numbers such that the two quadratic equations 19x^2+3x+2=0 and Ax^2+Bx+7=0 have a common root. What is the value of A+B

19x^2+3x+2=0 has roots

(-3±√143 i)/38

since both roots are complex, if the two quadratics share one root, they share both. So,

Ax^2+BX+7 must be a multiple of 19x^2+3x+2.

So, it must be 7/2 times, making it

19(7/2)x^2 + 3(7/2)x + 2(7/2)
= 66.5x^2 + 10.5x + 7

A+B = 77

thank you very much

To find the value of A + B, we need to determine the common root of the two quadratic equations:

1) 19x^2 + 3x + 2 = 0
2) Ax^2 + Bx + 7 = 0

To find the common root, we can equate the quadratic equations and solve for x. Here is how we do it:

1) Set the two quadratic equations equal to each other:
19x^2 + 3x + 2 = Ax^2 + Bx + 7

2) Rearrange the equation to set it equal to zero:
(A - 19)x^2 + (B - 3)x + 5 = 0

3) Since the two equations have a common root, they will have the same discriminant (B^2 - 4AC) and their quadratic coefficients (A - 19) and (B - 3) will be proportional (i.e., one will be a multiple of the other). This implies that their discriminants are equal, so we can set up an equation:

(B - 3)^2 - 4(A - 19)(5) = 0

4) Expand and simplify the equation:
B^2 - 6B + 9 - 20(A - 19) = 0
B^2 - 6B + 9 - 20A + 380 = 0
B^2 - 6B - 20A + 389 = 0

5) At this point, we have an equation with two variables (A and B) and one equation. We need another equation to solve for A and B. Looking at the initial equations, we see that the second quadratic equation has a constant term of 7. This implies that when x = 0, the equation evaluates to 7:

Ax^2 + Bx + 7 = 0
Putting x = 0:
7 = 0 + 0 + 7

Therefore, the common root is x = 0.

6) Substituting x = 0 into either of the quadratic equations, we can solve for B:
19(0)^2 + 3(0) + 2 = 0
0 + 0 + 2 = 0
2 = 0

This equation implies that 2 = 0, which is not true. Therefore, there is no common root, and the value of B is undefined.

As a result, A + B is also undefined.