Question

How do you simplify `sqrt(m+3) - sqrt (m-1) = 1`?

Answer 1

`color(green)(m=13/4)`

Explanation:

Given `sqrt(m+3)-sqrt(m-1)=1`

separating the roots:
`rarr color(white)("XX")sqrt(m+3)=sqrt(m-1)+1`

squaring both sides
`rarr color(white)("XX")m+3=(m-1) + 2sqrt(m-1) +1`

combining the `(-1)` and `+1`, and subtracting `m` from both sides
`rarr color(white)("XX")3=2sqrt(m-1)`

reversing the sides and dividing both sides by 2
`rarr color(white)("XX")sqrt(m-1)=3/2`

squaring both sides
`rarr color(white)("XX")m-1=9/4`

adding `1(=4/4)` to both sides
`rarr color(white)("XX")m=13/4`

Answer 2

`m=3 1/4`

Explanation:

`sqrt(m+3)-sqrt(m-1)=1`

`:.sqrt(m+3)=sqrt(m-1)+1`

square L.H.S.and R.H.S.

`:.(sqrt(m+3))^2=(sqrt(m-1)+1)^2`

`sqrta xx sqrta=a`

`:.m+3=(m-1)+2sqrt(m-1)+1`

`:.m+3=mcancel(-1)+2sqrt(m-1)cancel(+1)`

`:.m+3=m+2sqrt(m-1)`

`:.3=cancelmcancel(-m)+2sqrt(m-1)`

`:.3=2sqrt(m-1)`

`:.2sqrt(m-1)=3`

`:.sqrt(m-1)=3/2`

square L.H.S.and R.H.S.

`:.(sqrt(m-1))^2=(3/2)^2`

`:.m-1=9/4`

`:.m=1+9/4`

`:.m=13/4`

`:.m=3 1/4`

check:

`:.sqrt(3.25+3)-sqrt(3.25-1)=1`

`:.sqrt6.25-sqrt2.25=1`

`:.2.5-1.5=1`

`:.1=1`