1. ## Surds

Given that [a - 6 sqrt(5)] x [2 + b sqrt(5)] = -82, where a and b are rational numbers, find the possible values of a and b
[Ans = a = 4, b =3, or a = -45, b = -4/15]

I have expanded the expression to obtain
2a + sqrt 5 (ab-12) - 30 b = 82, but I don't know how to proceed to get simulataneous equations to solve for a and b.

Thanks

2. Originally Posted by Drdj
Given that [a - 6 sqrt(5)] x [2 + b sqrt(5)] = -82, where a and b are rational numbers, find the possible values of a and b
[Ans = a = 4, b =3, or a = -45, b = -4/15]

I have expanded the expression to obtain
2a + sqrt 5 (ab-12) - 30 b = 82, but I don't know how to proceed to get simulataneous equations to solve for a and b.

$\displaystyle (a - 6 \sqrt{5}) \cdot (2 + b \sqrt{5}) = 2a + (ab-12)\sqrt{5} - 30b = -82$
$\displaystyle 2a - 30b = -82$
$\displaystyle ab-12 = 0$
3. We need to know that "a and b are rational numbers" in order to be certain that $\displaystyle (ab- 12)\sqrt{5}$ is not a rational number and so we can so we must have $\displaystyle (ab- 12)\sqrt{5}= 0$.