jmc

Question

Let a and b be positive real numbers such that each of the equations x^2 + ax + 2b = 0 and x^2 + 2bx + a = 0 has real roots. Find the smallest possible value of a + b.

Accepted Answer

Since both quadratics have real roots, we must have

a^{2} \geq 8 b

and

4 b^{2} \geq 4 a,

or

b^{2} \geq a .

Then

b^{4} \geq a^{2} \geq 8 b .

Since

b > 0,

it follows that

b^{3} \geq 8,

so

b \geq 2.

Then

a^{2} \geq 16,

so

a \geq 4.

If

a = 4

and

b = 2,

then both discriminants are nonnegative, so the smallest possible value of

a + b

is

6 .