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Then Find the value of $s^4-18s^2-8s$

$$s=\sqrt a + \sqrt b +\sqrt c$$ $$s^2=a+b+c+2(\sqrt {ab} +\sqrt {bc} +\sqrt {ac})$$

I can’t seem to find a way around this obstacle. Squaring it again gives another $\sqrt {ab}+\sqrt {bc} +\sqrt {ac}$, and is seemingly never ending.

This drove me to the conclusion that the value of s cannot actually be found, and the question must be manipulated to get the required from. I don’t know how to do that though.

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How about evaluating the square of $\sqrt{ab}+\sqrt{bc}+\sqrt{ca}$ independently?

$$(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^2=ab+bc+ca+2\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c}) = 11+2s$$

so

$$\sqrt{ab}+\sqrt{bc}+\sqrt{ca}=\sqrt{11+2s}$$

From your work, we then get:

$$s^2=9+2\sqrt{11+2s}\Rightarrow s^2-9=2\sqrt{11+2s}\Rightarrow s^4-18s^2+81=44+8s$$

In conclusion $s^4-18s^2-8s=-37$.

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If we call:

$\rho=\sqrt{ab}+\sqrt{bc}+\sqrt{ca}$

we have, squaring s and using Vieta's formulas:

$s^2=9+2\rho$ [1]

but also, squaring $\rho$:

$\rho^2=11+2s$ [2]

Now take [1] as $s^2-9=2\rho$, square, substitute [2]... and it should be fine :)

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You should use the following properties of the roots of a cubic equation ax^3 + bx^2 + cx + d = 0:

a+b+c = -b/a = -(-9)/1 = 9

abc = -d/a = 1

and

ab + ac + bc = c/a = 11/1 =11

From these you can find: (a+b+c)^2 = 81 = a^2+b^2+c^2 + 2 (ab + ac + bc) = a^2+b^2+c^2 + 2 * 11 => a^2+b^2+c^2 = 81 - 22 = 59

Now expand s^2 and s^4 using these expressions and you should be one step closer.

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