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:
[tex]a + b = 18[/tex][tex]ab = 1[/tex]
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].
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:
[tex]a + b = 18[/tex][tex]ab = 1[/tex]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].