Question

The number of real roots of

(7 + 4$\sqrt{3}$)^{| x | – 8} + (7 – 4$\sqrt{3}$)^{| x | – 8} = 14 is -

• A. 0
• B. 2
• C. 4
• D. None of these

Answer 1

Answer: (C)

4

Answer 2

Note that 7 – $4\sqrt{3}$ = $\frac{1}{7 + 4\sqrt{3}}$

Putting y = (7 + 4$\sqrt{3}$)^{| x | – 8} , we get from (1)

y +$\frac{1}{y}$= 14 Ž y² – 14y + 1 = 0

Ž y = $\frac{14 \pm \sqrt{196 - 4}}{2}$= $\frac{14 \pm 8\sqrt{13}}{2}$

= 7 ± 4$\sqrt{3}$

= (7 + 4$\sqrt{3}$)^±1

Thus, (7 + 4$\sqrt{3}$)^{| x | – 8} = (7 + 4$\sqrt{3}$)^±1

Ž | x | – 8 = ± 1 Ž | x | = 9, 7

Ž x = ± 9, ± 7

\ there are four values of x satisfying the given equation.