`a ^(4) -12aa ^(2)-72=0`
Which of the following is the greatest possible value of a ?

Options

A. `sqrt(6+sqrt3)`
B. `sqrt(6(1+sqrt3))`
C. 12
D. `6(1+sqrt3)`

Answer

B

Explanation

Even thought the given equation contains the value `a ^(4),` this is a quadratic-style question. You can say that `x = a^(2), so x ^(2)-12x-72 =0.` To complete the square, restate this as `xd ^(2) -12x =72.` One-half of the x-coefficient is 6, which, when squared, becomes `36, `So,`x ^(2)-12x +36=108.` Factor to find that `(x -6) ^(2) = 108` and then take the square root of both sides to get `x -6 = pm sqrt(108)=36xx3,` the radical simplifies to `6 sqrt3.`
Now substitute `a ^(2)` back in for `x :a ^(2)-6 pm 6 sqrt3 and a ^(2) =6 pm 6 sqrt3.` Since the question asks for the root with the greatest value, you can ignore the root with the minus sign, `a ^(2) =6+ 6 sqrt3=6 (1 + sqrt3).` Take the square root of both sides, and ignore the root with the negative sign to get `a = sqrt(6(1+ sqrt3)).` (B) is correct.