A and B are real numbers such that the two quadratic equations 13x^2 + 3x +2 = 0 and Ax^2 + Bx + 5 = 0 have a common root. What is the value of A+B?
To find the value of A+B, we need to determine the common root of the two quadratic equations and then calculate the sum of the coefficients of x in the equation Ax^2 + Bx + 5 = 0.
Given two quadratic equations, 13x^2 + 3x + 2 = 0 and Ax^2 + Bx + 5 = 0, let's first find their common root.
For two quadratic equations to have a common root, their discriminants must be zero.
The discriminant of a quadratic equation Ax^2 + Bx + 5 = 0 can be calculated using the formula D = B^2 - 4AC, where A, B, and C are the coefficients of the equation.
Let's calculate the discriminant for the equation Ax^2 + Bx + 5 = 0:
Discriminant = B^2 - 4AC.
Since the coefficient of x^2 in the equation is A, the coefficient of x is B, and the constant is 5, we have:
Discriminant = B^2 - 4AC = B^2 - 4(A)(5) = B^2 - 20A.
To have a common root, both discriminants of the given quadratic equations should be zero:
For 13x^2 + 3x + 2 = 0, the discriminant is:
D1 = (3)^2 - 4(13)(2) = 9 - 104 = -95.
For Ax^2 + Bx + 5 = 0, the discriminant is:
D2 = B^2 - 4A(5) = B^2 - 20A.
Since D1 and D2 should be equal to zero, we have two equations:
D1 = -95 = 0,
D2 = B^2 - 20A = 0.
From the equation D1 = -95 = 0, we can see that -95 is not equal to zero, which means the equation 13x^2 + 3x + 2 = 0 does not have a real root.
Therefore, the two quadratic equations cannot have a common real root. Consequently, it is not possible to determine the value of A+B.