First slide

Theory of equations

Question

If $(7 - 4 \sqrt{3})^{x^{2} - 4 x + 3} + (7 + 4 \sqrt{3})^{x^{2} - 4 x + 3} = 14$ then the value of x is given by

Moderate

A

$2, 2 \pm \sqrt{2}$
B

$2 \pm \sqrt{3}, 3$
C

$3 \pm \sqrt{2}, 2$
D

None of these

Solution

since,
$(7 + 4 \sqrt{3}) (7 - 4 \sqrt{3}) = 1$
the given equation becomes
$\begin{aligned} y + \frac{1}{y} = 14 \\ where y = (7 - 4 \sqrt{3})^{x^{2} - 4 x + 3} \\ \Rightarrow y^{2} - 14 y + 1 = 0 \Rightarrow y = 7 \pm 4 \sqrt{3} \\ Now y = 7 \pm 4 \sqrt{3} \\ \Rightarrow x^{2} - 4 x + 3 = - 1 \\ \Rightarrow y = 7 - 4 \sqrt{3} \\ Also, x^{2} - 4 x + 3 = 1 \Rightarrow x = 2 \pm \sqrt{2} \end{aligned}$

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