If the quadratic equations, ax^2 + 2cx + b = 0 and ax^2 + 2bx + c = 0 (b not equals c) have a common root, then find the value of a + 4b + 4c

We have the equations ax^2 + 2cx + b = 0 and ax^2 + 2bx + c = 0 and its given that they have a common root. So assume that common roots is p.

Then we'll have

ap^2 + 2cp + b = 0

ap^2 + 2bp + c = 0

Subtracting these two equations, we'll get

2p(c - b) + (b - c) = 0

=> 2p(c - b) = (c - b)

Since c = b, (c - b) = 0, which means we can divide both the sides of the equation by (c - b). Doing so, we'll get,

=> 2p = 1

=> p = 1/2

Which means the common roots is 1/2. This means x = 1/2 will satisfy both the equations.

Now putting x = 1 in ax^2 + 2cx + b = 0, we'll get

a(1/2)^2 + 2c(1/2) + b = 0

=> a/4 + c + b = 0

=> (a + 4c + 4b)/4 = 0

=> a + 4b + 4c = 0

So, a + 4b + 4c is 0

*You may also notice that putting using x = 1/2 in ax^2 + 2bx + c = 0 would yield the same result