Hi,
Any advice with the problem below would be greatly appreciated.
X = √a + 1/√a + 1; X>0; express X - 2X in terms of a.
btw, answer = a + 1 + 1/a; how do you derive it though?
Thank you,
Sean.
Hello, Seaniboy!
$\displaystyle x \:=\:\sqrt{a} + \tfrac{1}{\sqrt{a}} + 1,\;x > 0.$
$\displaystyle \text{Express }x{\color{red}^2}-2x\text{ in terms of }a.$
$\displaystyle \text{Answer: }\:a + 1 + \tfrac{1}{a}$
$\displaystyle x^2-2x \;=\;x(x-2)$
. . . . . . $\displaystyle =\;\left(\sqrt{a} + \tfrac{1}{\sqrt{a}} + 1\right)\left(\sqrt{a} + \tfrac{1}{\sqrt{a}} + 1 - 2\right)$
. . . . . . $\displaystyle =\;\bigg(\left[\sqrt{a} + \tfrac{1}{\sqrt{a}}\right] + 1\bigg)\bigg(\left[\sqrt{a} + \tfrac{1}{\sqrt{a}}\right] - 1\bigg)$
. . . . . . $\displaystyle =\;\left(\sqrt{a} + \tfrac{1}{\sqrt{a}}\right)^2 - 1^2$
. . . . . . $\displaystyle =\;a + 2 + \tfrac{1}{a} - 1$
. . . . . . $\displaystyle =\;a + 1 + \tfrac{1}{a}$