If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist?

Question

If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist?

• One
• Two
• Three
• Infinitely many

Also, what are these pairs?

Answer 1

Since $a, b$ are zeros, they each satisfy the equation:

$$x=a: \quad a^2 + a\cdot a + b = 0\iff 2a^2 + b = 0 \iff b = -2a^2$$

$$x = b:\quad b^2 + ab + b = 0$$

Substituting $b = -2a^2$ into the second equation: $$(-2a^2)^2 + a(-2a^2) + -2a^2 = 0 \iff 4a^4 -2a^3 - 2a^2 = 2a^2(2a^2 -a - 1) = 0$$

The right-hand factor itself factors nicely, giving you 3 solutions to $a$, and then you can find the corresponding value for $b$ by using $b = -2a^2$.

Answer 2

Sum of the roots is $a+b = -a \implies 2a+b = 0$.

Product of the roots is $ab = b \implies b = 0 \text{ or } a = 1$.

If $b=0$, then $2a+ 0 = 0 \implies a = 0$.

If $a=1$, then $2+b = 0 \implies b = -2$.

Answer 3

Hint: if $a$ and $b$ are the zeroes then $x^{2}+ax+b=\left(x-a\right)\left(x-b\right)$.