Question

How do you solve `sqrt(m-1) + 2 = m-5`?

Answer 1

`m=10`

Explanation:

Given:

`sqrt(m-1)+2 = m-5`

First isolate the square root on one side by subtracting `2` from both sides to get:

`sqrt(m-1) = m-7`

Square both sides. Note that the resulting equation is the same as that which you would get by squaring `-sqrt(m-1) = m-7`. So this will add a spurious solution that we need to eliminate later.

`m-1 = (m-7)^2 = m^2-14m+49`

Subtract `m-1` from both ends to get:

`0 = m^2-15m+50 = (m-5)(m-10)`

So:

`m=5" "` or `" "m=10`

Trying `m=5`, we find:

`sqrt(color(blue)(5)-1)+2 = sqrt(4)+2 = 2+2 = 4 != 0 = color(blue)(5)-5`

So `m=5` is not a solution of the original equation.

Trying `m=10`, we find:

`sqrt(color(blue)(10)-1)+2 = sqrt(9)+2 = 3+2 = 5 = color(blue)(10)-5`

So `m=10` is a solution of the original equation.