Quadratic Equation Question 1
Question 1 - 2024 (01 Feb Shift 1)
Let $S={x \in R:(\sqrt{3}+\sqrt{2})^{x}+(\sqrt{3}-\sqrt{2})^{x}=10 }$.
Then the number of elements in $S$ is :
(1) 4
(2) 0
(3) 2
(4) 1
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Answer (3)
Solution
$(\sqrt{3}+\sqrt{2})^{x}+(\sqrt{3}-\sqrt{2})^{x}=10$
Let $(\sqrt{3}+\sqrt{2})^{x}=t$
$t+\frac{1}{t}=10$
$t^{2}-10 t+1=0$
$t=\frac{10 \pm \sqrt{100-4}}{2}=5 \pm 2 \sqrt{6}$
$(\sqrt{3}+\sqrt{2})^{x}=(\sqrt{3} \pm \sqrt{2})^{2}$
$x=2$ or $x=-2$
Number of solutions $=2$
