Grade 11Algebra

find the solutions of equation

(x-1/x)^1/2 + (1-1/x)^1/2 = x

ankit agrawal

14 Years agoGrade 11

2 Answers

Shashwat Tiwari

Approved Tutor Answer14 Years ago

(x-1/x)^1/2 = x-(1-1/x)^1/2

squaring both the sides..........

x-1/x = x² + 1 - 1/x - 2(x²-x)^1/2...................

x²-x+1=2(x²-x)^1/2.....................

now, put x²-x=t.......

t+1=2(t)^1/2

again square both sides.............

(t-1)²=0

t=1

Hence, x²-x=1

x=(1±(5)^1/2)/2..............

jagdish singh singh

Approved Tutor Answer14 Years ago

$\hspace{-16}\mathbf{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}=x}$\\\\\\ Let $\mathbf{\sqrt{x-\frac{1}{x}}=a}$ and $\mathbf{\sqrt{1-\frac{1}{x}}=b}$\\\\\\ $\mathbf{a+b=x........................................(1)}$\\\\\\ $\mathbf{a^2-b^2=x-1\Leftrightarrow (a+b).(a-b)=(x-1)}$\\\\\\ $\mathbf{a-b=\frac{x-1}{x}..................................(2)}$\\\\\\ $\mathbf{2a=x+\frac{x-1}{x}\Leftrightarrow =1+a^2\Leftrightarrow }$\\\\\\ $\mathbf{a^2-2a+1=0\Leftrightarrow (a-1)^2=0\Leftrightarrow a=1}$\\\\\\ So $\mathbf{\sqrt{x-\frac{1}{x}}=a=1\Leftrightarrow x-\frac{1}{x}=1}$\\\\\\ So $\mathbf{x=\frac{-1+\pm 5}{2}\Leftrightarrow \boxed{\bold{x=\frac{1+\sqrt{5}}{2}}}}$$