Full solution

Q. e)

\frac{1}{\sqrt{x-1}}+\sqrt{x-1}=2

Isolate square root term: First, let's isolate the square root term on one side of the equation. To do this, we subtract $\sqrt{x-1}$ from both sides. $\newline$ Calculation: $\frac{1}{\sqrt{x-1}} = 2 - \sqrt{x-1}$
Eliminate fraction: Now, to get rid of the fraction, we can multiply both sides by $\sqrt{x-1}$ . $\newline$ Calculation: $1 = (2 - \sqrt{x-1}) \cdot \sqrt{x-1}$
Distribute square root: Next, we'll distribute the $\sqrt{x-1}$ on the right side of the equation. $\newline$ Calculation: $1 = 2\sqrt{x-1} - (\sqrt{x-1})^2$
Simplify equation: Simplify the equation by squaring the square root term and moving all terms to one side to set the equation to zero. $\newline$ Calculation: $(\sqrt{x-1})^2 = x-1$ $\newline$ So, $1 = 2\sqrt{x-1} - (x-1)$
Move terms to one side: Add $x - 1$ to both sides to get all $x$ terms on one side and constant terms on the other. $\newline$ Calculation: $x - 1 + 1 = 2\sqrt{x-1}$
Solve for $x$ : Simplify the left side of the equation. $\newline$ Calculation: $x = 2\sqrt{x-1}$
Square both sides: Now, to solve for $x$ , we square both sides of the equation to eliminate the square root. $\newline$ Calculation: $x^2 = (2\sqrt{x-1})^2$
Square right side: Square the right side of the equation. $\newline$ Calculation: $x^2 = 4 \times (x-1)$
Distribute $4$ : Distribute the $4$ on the right side of the equation. $\newline$ Calculation: $x^2 = 4x - 4$
Set equation to zero: Move all terms to one side to set the equation to zero and solve for $x$ . $\newline$ Calculation: $x^2 - 4x + 4 = 0$
Factor quadratic equation: Factor the quadratic equation. $\newline$ Calculation: $(x - 2)(x - 2) = 0$
Set factors equal: Set each factor equal to zero and solve for $x$ . $\newline$ Calculation: $x - 2 = 0$ $\newline$ So, $x = 2$