Question

How does `t^2-2t-1=0` become `t=1+-sqrt2`?

Answer 1

`"see explanation"`

Explanation:

`"solve for t using the method of "color(blue)"completing the square"`

`• " the coefficient of the "t^2" term must be 1 which it is"`

`• " add "(1/2"coefficient of the t- term ")^2" to both sides"`

`"add 1 to both sides"`

`t^2-2t=1`

`t^2+2(-1)t color(red)(+1)=1color(red)(+1)`

`(t-1)^2=2`

`color(blue)"take the square root of both sides"`

`sqrt((t-1)^2)=+-sqrt2larrcolor(blue)"note plus or minus"`

`t-1=+-sqrt2`

`"add 1 to both sides to obtain"`

`t=1+-sqrt2`

Answer 2

An alternate way using the quadratic formula:

Explanation:

Another way to find solutions for `t` is to use the quadratic formula:

`t = (-b \pm sqrt(b^2-4ac)) / (2a)`

with `a=1, b=-2, c=-1`

`t = (2 \pm sqrt((-2)^2-4(1)(-1))) / (2(1))`

`t = (2 \pm sqrt(4+4)) / 2`

`t = (2 \pm sqrt8) / 2`

`t = (2 \pm 2sqrt2) / 2`

`t = 1 \pm sqrt2`