Radical Equations...?

Question

I know how to work with Radical Equations. I just need some help with this one:

It is asking for the area of a Rhombus, with the equation given:

Area = 1/2(D1)(D2)

D1 being √x + 2
D2 being √x - 1

The area is said to be 9, so I know that when you plug in the values given the final equation should look like this:

9 = 1/2(√x + 2)(√x - 1)

The answers are said to be 3 and 6. I am stuck on what to do next. Should I act as if I am using two binomials? (√x + 2)(√x -1) = x - 1√x + 2√x - 2, which equals x + 1√x - 2.

Can anyone offer some advice that will lead me to the correct path?

Answer 1

9=1/2(x^2+x-2)^1/2
square
81=1/4(x^2+x-2)
324=x^2+x-2
x^2+x-326=0
solve

Answer 2

Firstly, you have : (1/2)(√x + 2)(√x - 1) = 9

Multiply through by 2 to get rid of fractions :
(√x + 2)(√x - 1) = 18

Multiplying out, as you have done correctly, gives :
x + √x - 2 = 18

Now add 2 to both sides :
x + √x = 20

Now you need to get √x on its own,
so you can square it.
Subtracting x from both sides gives :
√x = 20 - x

Now you can square both sides, gather up all
terms on one side to make a quadratic equation
and solve for x by factorising. Then plug your
values for x into √x + 2 and √x - 1.
Do this now before reading any further. If you
have any troubles the rest of the answer follows.
--------------------
Squaring both sides gives :
x = 400 - 40x + x^2

Subtracting x from both sides gives :
x^2 - 41x + 400 = 0

Factorising gives :
(x - 16)(x - 25) = 0

Therefore, x = 16 or 25
and so, √x = ±4 or ±5

Now we can discard the negative values for √x,
because D1 and D2 are both positive lengths.

Thus, if √x = 4, then √x + 2 = 6 and √x -1 = 3.
Also, if √x = 5, then √x + 2 = 7 and √x - 1 = 4.

Radical problems must always be checked, because
you may get contradictory results.
Notice when √x = 5, (1/2)(D1)D2) = (1/2)(7)(4) = 14
which is certainly not equal to 9, so therefore,
this is not a legitimate solution.

Perversely, the other legitimate solution is when
√x = -5, which had to be discarded.
Note that here, √x + 2 = -3 and √x - 1 = -6,
and (1/2)(-3)(-6) does equal 9.

So the answers are : D1 = 6 and D2 = 3.