Find the number of ordered pairs $(a,b)$ of real numbers such that $\bullet$ $a$ is a root of $x^2 + ax + b = 0.$ $\bullet$ $b$ is a root

deleted.

If a is a root, then

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

2a^2 + b = 0

b = -2a^2    (1)

If b is a root, then

b^2 + ab + b = 0    (2)

Sub (1) into (2)

(-2a^2)^2 +a(-2a^2) + (-2a^2) = 0    simplify

4a^4 -2a^3 - 2a^2 = 0   factor

(2a^2) (2a^2 - a - 1) = 0

Set both factors to 0 and solve

2a^2 = 0        (2a + 1) (a - 1) = 0

a = 0                a = -1/2       a = 1

Using (1).....

When a = 0, b = 0     (trivial solution)

When a = -1/2 , b = -1/2

When  a = 1, b = -2

Proof

x^2 + 0x +0 = 0  .......has the root x = 0.....(a,b) = (0,0)

x^2 - (1/2)x - 1/2 = 0     multiply through by 2

2x^2  - x^2 - 1  =  0  factor

(2x + 1) (x - 1) = 0

Has the roots  x = -1/2  and x  = 1 

So  .....(a,b) = (-1/2, -1/2)

x^2 -2x + 1 = 0      factor

(x -1)  (x -1) = 0.......so x = 1 is a root......(a, b) = (1, -2)

So......

(a, b)  = (0, 0)    (trivial)

(a, b) = (-1/2, -1/2)

(a, b) = (1, -2)