Question

How do you solve `|x^2-4x-5|=7`?

Answer 1

There are 4 solution for `x` because there are two quadratics. Each having two answers:

`"quadratic 1" ->x" "=" " 2+sqrt(2)"; "2-sqrt(2)"; "`
`"quadratic 2" ->x" "=" "6"; "-2`

Explanation:

what is inside the | | ( an absolute value) is always positive.

But this value when made positive is 7.

That means that what is inside the | | can be either of `+-7`

So we have two conditions
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Condition 1")`

`x^2-4x-5=-7`

`x^2-4x+2=0`

Completing the square:

`(x-2)^2+2-4=0`
`(x-2)^2-2=0`

`x=2+-sqrt(2) larr" both of these work when checked"`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Condition 2")`

`x^2-4x-5=+7`

`x^2-4x-12=0`

Completing the square

`(x-2)^2-12-4=0`
`(x-2)^2-16=0`

`x=2+-4 larr" both of these work when checked"`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Thus `x= 2+sqrt(2)"; "2-sqrt(2)"; "6"; "-2`