[tex]( \sqrt{5} - 2)^{x} + ( \sqrt{5} + 2) ^{x} = 18[/tex]

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вопрос

To simplify this equation, let's substitute [tex]a = (\sqrt{5} - 2)^x [/tex] and [tex]b = (\sqrt{5} + 2)^x[/tex].

Now we have [tex]a + b = 18[/tex].

We know that [tex](\sqrt{5} - 2)(\sqrt{5} + 2) = 1[/tex] because [tex](a - b)(a + b) = a^2 - b^2[/tex].

This implies that [tex]ab = 1[/tex].

So, now we have the system of equations:

From the second equation, we have [tex]b = \frac{1}{a}[/tex].

Substitute this expression into the first equation:
[tex]a + \frac{1}{a} = 18[/tex].

Multiplying by [tex]a[/tex] to clear the denominator gives:
[tex]a^2 + 1 = 18a[/tex].

Rearranging gives:
[tex]a^2 - 18a + 1 = 0[/tex].

This is a quadratic equation in [tex]a[/tex] which can be solved using the quadratic formula. The solutions will give values for [tex]a[/tex] and [tex]b[/tex].