First slide

Theory of equations

Question

Roots of the equation $(a + \sqrt{b})^{x^{2} - 15} + (a - \sqrt{b})^{x^{2} - 15} = 2 a,$ where $a^{2} - b = 1$ , are

Moderate

A

$\pm 4$
B

$\pm 3$
C

$\pm \sqrt{14}$
D

$\pm \sqrt{5}$

Solution

we have, $(a + \sqrt{b})^{x^{2} - 15} + \frac{1}{(a + \sqrt{b})^{x^{2} - 15}} = 2 a$
$\begin{array}{ll} y^{2} - 2 ay + 1 = 0, where y = (a + \sqrt{b})^{x^{2} - 15} \\ ∴ & y = a \pm \sqrt{a^{2} - 1} = a \pm \sqrt{b} = (a + \sqrt{b})^{x^{2} - 15} \\ So, & x^{2} - 15 = \pm 1 \\ ∴ & x = \pm 4, \pm \sqrt{14} \end{array}$

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