Q. 339 - Intermediate Algebra

Question

In the following exercises, solve.
$\sqrt{2x-1}-\sqrt{x-1}=1$

Step-by-Step Solution

Answer

The solutions of the given equation are $x=1$ and $x=5$.

Step 1. Given information.

The given equation is:

$\sqrt{2x-1}-\sqrt{x-1}=1$

Step 2. Simplify the equation.

The given equation can be written as:

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

Take square on both sides.

$\begin{array}{rcl}(\sqrt{2x-1}{)}^2& =& (1+\sqrt{x-1}{)}^2\\ 2x-1& =& 1+2\sqrt{x-1}+(x-1)\\ 2x-1-x& =& 2\sqrt{x-1}\\ x-1& =& 2\sqrt{x-1}\end{array}$

Step 3. Solve the equation.

Square both sides.

$\begin{array}{rcl}(x-1{)}^2& =& 4(x-1)\\ (x-1{)}^2-4(x-1)& =& 0\\ (x-1)(x-1-4)& =& 0\\ (x-1)(x-5)& =& 0\\ x& =& 1,5\end{array}$

Step 4. Conclusion.

The solutions of the given equation are $x=1$ and $x=5$.