If \( {x }^{ 2} \) + x = 19
find \( {(x + 5) }^{ 2} \) + \(\frac{1}{(x+5)^2}\)
Answer & explanation
Correct answer: option 2
Let x + 5 = t
x = t - 5
Put in \( {x }^{ 2} \) + x = 19
⇒ (t - 5)2 + t - 5 = 19
⇒ t2 + 25 - 10t + t - 5 = 19
⇒ t2 - 9t + 1 = 0
⇒ t + \(\frac{1}{t}\) = 9
⇒ t2 +\(\frac{1}{ {t }^{2 }}\) = 92 - 2 = 79