Question

The number of real solutions of the equation $(3+2\sqrt{2}{)}^{x^2-3}+(3-2\sqrt{2}{)}^{x^2-3}=6$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is D $4$

$(3+2\sqrt{2}{)}^{x^2-3}+(3-2\sqrt{2}{)}^{x^2-3}=6$
We know that,
$3-2\sqrt{2}=\frac{1}{3+2\sqrt{2}}$
Let $(3+2\sqrt{2}{)}^{x^2-3}=a$, then given equation is
$a+\frac{1}{a}=6\Rightarrow a^2-6a+1=0\Rightarrow a=\frac{6\pm \sqrt{32}}{2}\Rightarrow a=3\pm 2\sqrt{2}$

When $a=3+2\sqrt{2}$
$(3+2\sqrt{2}{)}^{x^2-3}=3+2\sqrt{2}\Rightarrow x^2-3=1\Rightarrow x=\pm 2$

When $a=3-2\sqrt{2}$
$(3+2\sqrt{2}{)}^{x^2-3}=3-2\sqrt{2}\Rightarrow x^2-3=-1\Rightarrow x=\pm \sqrt{2}$

Hence, the given has $4$ real solutions.