Rearranging Formulas Involving Square Roots

[nortski]

When rearranging this formula involving a square root on the right side, can someone please explain to me where the +6m in the second step comes from when squaring the left side? I'm going through my revision guide and I don't feel this has been explained at all, although I am probably missing something simple.

Please excuse the crudely put together image taken from the guide as I don't know how to write this formula as text.

 

[MarkFL]

When we square a binomial, the result is:

[MATH](x+y)^2=x^2+2xy+y^2[/MATH]
In this case, we have:

[MATH](m+3)^2=m^2+2(m)(3)+3^2=m^2+6m+9[/MATH]

 

[nortski]

Ah so it's the same as (m + 3)(m + 3)?

 

[JeffM]

They left out some steps.

[MATH]\{2(m + 3\}^2 = (\sqrt{n + 5})^2 = n + 5 \implies[/MATH]
[MATH]2^2 * (m + 3)^2 = n + 5 \implies[/MATH]
[MATH]4\{(m + 3) * (m + 3)\} = n + 5 \implies[/MATH]
[MATH]4\{m(m + 3) + 3(m + 3)\} = n + 5 \implies [/MATH]
[MATH]4(m^2 + 3m + 3m + 9) = n + 5 \implies[/MATH]
[MATH]4(m^2 + 6m + 9) = n + 5 \implies[/MATH]
[MATH]4m^2 + 24m + 36 = n + 5 \implies[/MATH]
[MATH]n = 4m^2 + 24m + 31.[/MATH]

 

[Steven G]

Ah so it's the same as (m + 3)(m + 3)?

Yes! The rule is that [A*B]ⁿ=Aⁿ*Bⁿ.

Let' look at this for n=3: [A*B]³ = (AB)(AB)(AB) = ABABAB = AAABBB= A³*B³.

In your example A is 2, B is (m+3) and n is 2.