The values of x which satisfy the expression \((5+2\sqrt6)^{x^2+3}\)

1.	The values of x which satisfy the expression \((5+2\sqrt6)^{x^2+3}\) + \((5-2\sqrt6)^{x^2-3}\)= 10 are :(a) ± 2, ± √3 (b) ± √2, ± 4 (c) ± 2, ± √2 (d) 2, √2, √3
Answer» (c) ± 2, ± √2 Let y = 5 + 2√6. Then \(\frac{1}{y}\) = 5 - 2√6. Thus the given expression reduces to \(y^{x^2-3}\) + \(\big(\frac{1}{y}\big)^{x^2-3}\) = 10 Again let \(y^{x^2-3}\) = t. Then, t + \(\frac{1}{t}\) = 10 ⇒ t² - 10t + 1 = 0 ⇒ t = \(\frac{10±\sqrt{100-4}}{2}\) = \(\frac{10 ±\sqrt{96}}{2}\) = \(\frac{10 ±4\sqrt{6}}{2}\) = 5 ± 2√6 \((5+2\sqrt6)^{x^2-3}\) = 5 ± 2√6 = (5 ± 2√6)^±1 ⇒ x² – 3 = 1 or x² – 3 = – 1 ⇒ x² = 4 or x² = 2 ⇒ x = ± 2 or x = ± √2

The values of x which satisfy the expression \((5+2\sqrt6)^{x^2+3}\) + \((5-2\sqrt6)^{x^2-3}\)= 10 are :(a) ± 2, ± √3 (b) ± √2, ± 4 (c) ± 2, ± √2 (d) 2, √2, √3

Answer»

(c) ± 2, ± √2

Let y = 5 + 2√6. Then \(\frac{1}{y}\) = 5 - 2√6.

Thus the given expression reduces to \(y^{x^2-3}\) + \(\big(\frac{1}{y}\big)^{x^2-3}\) = 10

Again let \(y^{x^2-3}\) = t. Then,

t + \(\frac{1}{t}\) = 10 ⇒ t² - 10t + 1 = 0

⇒ t = \(\frac{10±\sqrt{100-4}}{2}\) = \(\frac{10 ±\sqrt{96}}{2}\)

= \(\frac{10 ±4\sqrt{6}}{2}\) = 5 ± 2√6

\((5+2\sqrt6)^{x^2-3}\) = 5 ± 2√6 = (5 ± 2√6)^±1

⇒ x² – 3 = 1 or x² – 3 = – 1

⇒ x² = 4 or x² = 2

⇒ x = ± 2 or x = ± √2