91Ó°ÊÓ

The inequality in standard form is $-x^2+2x+1â‰\yen 0$

Change the inequality sign to an equal sign and then solve the equation.

$-x^2+2x+1=0$

Use the quadratic formula to solve the equation.

$x=\frac{-bÂ\pm \sqrt{b^2-4ac}}{2a}$

Compare the quadratic equation with $a{x}^2+bx+c=0$

Here,

$a=-1,b=2,c=1$

The solution of the quadratic equation is,

role="math" localid="1645594321413" $$\begin{gathered} x=\frac{-\left(2\right)Â\pm \sqrt{{\left(2\right)}^2-4\left(-1\right)\left(1\right)}}{2\left(-1\right)} \\ =\frac{-2Â\pm \sqrt{4+4}}{-2} \\ =\frac{-2Â\pm \sqrt{8}}{-2} \end{gathered}$$

Simplify the radicand.
role="math" localid="1645594367647" $x=\frac{-2Â\pm 2\sqrt{2}}{-2}$

Remove the common factor, 2.

$$\begin{gathered} x=\frac{-1Â\pm \sqrt{2}}{-1} \\ x=\sqrt{2}+1,1-\sqrt{2} \end{gathered}$$

Use the critical points $1-\sqrt{2}$and $1+\sqrt{2}$to divide the number line into intervals.

The above number line shows the sign of each quadratic expression using test points from each interval.

Take test points$x=-1,x=0,x=3$

Check the inequality sign for the above test points.

For $x=-1$,

$$\begin{gathered} -x^2+2x+1=-{\left(-1\right)}^2+2\left(-1\right)+1 \\ =-1-2+1 \\ =-2 \end{gathered}$$

For $x=0$,

role="math" localid="1645596900872" $$\begin{gathered} -x^2+2x+1=-{\left(0\right)}^2+2\left(0\right)+1 \\ =0+0+1 \\ =1 \end{gathered}$$

For $x=3$,

$$\begin{gathered} -x^2+2x+1=-{\left(3\right)}^2+2\left(3\right)+1 \\ =-9+6+1 \\ =-2 \end{gathered}$$

The expression equals 0 at the points $1-\sqrt{2}$and $1+\sqrt{2}$

The inequality is positive in the second interval only.

The solution, in interval notation, is $\left[1-\sqrt{2},1+\sqrt{2}\right]$