Sat math completing the square

Test Preparation
1

P = 1/3s^2-5s+18

What khan academy says is

P= 1/3(s^2-15s)+18
= 1/3(s-15/2)^2-1/3(15/2)^2+18

The rest i understand but the second line
It isn’t like any normal completing the square thing

The middle is the same you did -15/2 then square them to add both side but why add to the s^2 thing . Then repeating -1/3(15/2)^2

Q2 h(t) = -16t^2 +64t+4
= -16( t^2-4t) +4

Then this the most confused part

Where does -64 come from ?

h(t) -64 = -16(t^2-4t+4) +4

And the another +4 that add to middle part ?

Thank you
Adele

2

Hmm I do not understand that method. To find zeros, constants should be isolated to one side and variables to the other in order to create a perfect square trinomial.

Could you provide some context like if you wanted to find zeros or a vertex because I doubt that the question just asked you to complete the square willy nilly (if it did, this will not be on the SAT).

3

@Adeleloveshiro For the first one, it seems like they are completing the square for the expression s^2 - 15s:

s^2 - 15s = (s - 15/2)^2 - (15/2)^2

You can verify that the above two expressions are equal.

We have the equation

h(t) = -16(t^2 - 4t) + 4

Now t^2 - 4t + 4 is the square of a polynomial (t - 2). What they did was subtract 64 from both sides and rewrite the right-hand side as -16(t^2 - 4t + 4) + 4 (note that -16*4 = -64). You can also verify that the two equations are equivalent.