Question

Suppose a, b are integers and a + b is a root of x2 + ax + b = 0. What is the maximum possible value of b2

Answer 1

Given: a, b are integers and a + b is a root of x2 + ax + b = 0

To find: Maximum possible value of b^2.

Solution:

• Now we have given the root is a + b, so putting this in the equation, we get:

              (a + b)^2 + a(a + b) + b = 0

              a^2 + b^2 + 2ab + a^2 + ab + b = 0

              b^2 + (3a + 1)b + 2a^2 = 0

• Now solving using formula, we get:

             -(3a + 1) ± √(3a + 1)^2 - 4(2a^2) / 2

             -(3a + 1) ± √a² + 6a + 1 / 2

• As, a,b ∈ a² + 6a + 1 ,  so a² + 6a + 1 must be a perfect square.

             a(6 + a) + 1

• Possible value of a are 0 or -6.

• Now, if a = 0, then b = -1 or 0

• If a = -6, then b =9 or 8

• So maximum value of b will be 81(square of 9)

Answer:

          So the maximum value of b will be 81.