Question

What is the positive solution to the equation 0 = –x2 + 2x + 1?

Answer 1

The equation $0=-x^2+2x+1$ is equivalent to the equation $x^2-2x-1=0.$

Solve this equation:

1. $D=b^2-4ac=(-2)^2-4\cdot 1\cdot (-1)=4+4=8.$

2. $\sqrt{D}=\sqrt{8}=2\sqrt{2}.$

3.

$x_1=\dfrac{-b+\sqrt{D}}{2a}=\dfrac{2+2\sqrt{2}}{2}=1+\sqrt{2}$

and

$x_2=\dfrac{-b-\sqrt{D}}{2a}=\dfrac{2-2\sqrt{2}}{2}=1-\sqrt{2}.$

4. Since $\sqrt{2}>1,$ you have that $1-\sqrt{2}$

Answer: the positive solution is $x=1+\sqrt{2}.$