Chapter A.5, Problem 65E

To determine

Tofind:The solution of the equation $x^2-2x-1=0$ .

Answer to Problem 65E

The solutions of the equation $x^2-2x-1=0$ are $x=1+\sqrt{2}$ and $x=1-\sqrt{2}$ .

Explanation of Solution

Given information:

The given equation is $x^2-2x-1=0$ .

Calculation:

Solve the given equation.

  $\begin{array}{c}{x}^2-2x-1=0\\ x^2-2\cdot 1\cdot x=1\\ x^2-2\cdot 1\cdot x+1^2=1+1^2\left\{\text{Add}{1}^2\text{both}\text{the}\text{sides}\right\}\\ {\left(x-1\right)}^2=1+1\left\{\begin{array}{l}\text{We}\text{know}\text{that}\\ {\left(a-b\right)}^2=a^2-2ab+b^2\end{array}\right\}\end{array}$

Further simplify the equation.

  $\begin{array}{c}{\left(x-1\right)}^2=1+1\\ x-1=\pm \sqrt{2}\\ x=1\pm \sqrt{2}\end{array}$

Since, there are two possible values of $x$ . Solveeach.

  $x=1+\sqrt{2}$

and

  $x=1-\sqrt{2}$

From the above, it is clear that the values of $x$ are $1+\sqrt{2}$ and $1-\sqrt{2}$ .

Therefore, the solutions of the equation $x^2-2x-1=0$ are $x=1+\sqrt{2}$ and $x=1-\sqrt{2}$ .