(15+414)x+(15−414)x=30

We have,(15+414)x+(15−414)x=30⇒ (15+414)x+1(15+414)x=30⇒ a+1a=30, where a=15+414Also, a2−30a+1=0⇒ a=30±900−42=30±8142=15±414When a=15+414⇒ (15+414)x=(15+414)⇒ x=1When a=(15−414)⇒ (15+414)x=(15−414)⇒ (15+414)x=(15+414)−1⇒ x=−1Hence the solution set is {-1, 1}.

(15+414)x+(15−414)x=30

We have,(15+414)x+(15−414)x=30⇒ (15+414)x+1(15+414)x=30⇒ a+1a=30, where a=15+414Also, a2−30a+1=0⇒ a=30±900−42=30±8142=15±414When a=15+414⇒ (15+414)x=(15+414)⇒ x=1When a=(15−414)⇒ (15+414)x=(15−414)⇒ (15+414)x=(15+414)−1⇒ x=−1Hence the solution set is {-1, 1}.

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$(15 + 4 \sqrt{14})^{x} + (15 - 4 \sqrt{14})^{x} = 30$

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{-11, 1}

{0, 1}

{-1, 1}

answer is C.

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Detailed Solution

We have,

$\begin{array}{l} (15 + 4 \sqrt{14})^{x} + (15 - 4 \sqrt{14})^{x} = 30 \\ \Rightarrow (15 + 4 \sqrt{14})^{x} + \frac{1}{(15 + 4 \sqrt{14})^{x}} = 30 \end{array}$

$\Rightarrow a + \frac{1}{a} = 30$ , where $a = 15 + 4 \sqrt{14}$

Also, $a^{2} - 30 a + 1 = 0$

$\begin{array}{l} \Rightarrow a = \frac{30 \pm \sqrt{900 - 4}}{2} \\ = \frac{30 \pm 8 \sqrt{14}}{2} = 15 \pm 4 \sqrt{14} \end{array}$

When $a = 15 + 4 \sqrt{14}$

$\begin{array}{l} \Rightarrow (15 + 4 \sqrt{14})^{x} = (15 + 4 \sqrt{14}) \\ \Rightarrow x = 1 \end{array}$

When $a = (15 - 4 \sqrt{14})$

$\begin{array}{l} \Rightarrow (15 + 4 \sqrt{14})^{x} = (15 - 4 \sqrt{14}) \\ \Rightarrow (15 + 4 \sqrt{14})^{x} = (15 + 4 \sqrt{14})^{- 1} \\ \Rightarrow x = - 1 \end{array}$