Solve for: x^2-2x=1

Expression: ${x}^{2}-2x=1$

Write the quadratic equation in standard form

${x}^{2}-2x-1=0$

Identify the coefficients $a$, $b$ and $c$ of the quadratic equation

$\begin{array} { l }a=1,& b=-2,& c=-1\end{array}$

Substitute $a=1$, $b=-2$ and $c=-1$ into the quadratic formula $x=\frac{ -b\pm\sqrt{ {b}^{2}-4ac } }{ 2a }$

$x=\frac{ -\left( -2 \right)\pm\sqrt{ {\left( -2 \right)}^{2}-4 \times 1 \times \left( -1 \right) } }{ 2 \times 1 }$

Any expression multiplied by $1$ remains the same

$x=\frac{ -\left( -2 \right)\pm\sqrt{ {\left( -2 \right)}^{2}-4 \times \left( -1 \right) } }{ 2 \times 1 }$

Any expression multiplied by $1$ remains the same

$x=\frac{ -\left( -2 \right)\pm\sqrt{ {\left( -2 \right)}^{2}-4 \times \left( -1 \right) } }{ 2 }$

When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses

$x=\frac{ 2\pm\sqrt{ {\left( -2 \right)}^{2}-4 \times \left( -1 \right) } }{ 2 }$

Evaluate the power

$x=\frac{ 2\pm\sqrt{ 4-4 \times \left( -1 \right) } }{ 2 }$

Any expression multiplied by $-1$ equals its opposite

$x=\frac{ 2\pm\sqrt{ 4+4 } }{ 2 }$

Add the numbers

$x=\frac{ 2\pm\sqrt{ 8 } }{ 2 }$

Simplify the radical expression

$x=\frac{ 2\pm2\sqrt{ 2 } }{ 2 }$

Write the solutions, one with a $+$ sign and one with a $-$ sign

$\begin{array} { l }x=\frac{ 2+2\sqrt{ 2 } }{ 2 },\\x=\frac{ 2-2\sqrt{ 2 } }{ 2 }\end{array}$

Simplify the expression

$\begin{array} { l }x=1+\sqrt{ 2 },\\x=\frac{ 2-2\sqrt{ 2 } }{ 2 }\end{array}$

Simplify the expression

$\begin{array} { l }x=1+\sqrt{ 2 },\\x=1-\sqrt{ 2 }\end{array}$

The equation has $2$ solutions

$\begin{align*}&\begin{array} { l }x_1=1-\sqrt{ 2 },& x_2=1+\sqrt{ 2 }\end{array} \\&\begin{array} { l }x_1\approx-0.414214,& x_2\approx2.41421\end{array}\end{align*}$