Question

How do you solve `v^ { 2} - 6v = 35` by completing the square?

Answer 1

By completing the square, we see that

`v^2-6v-35=v^2-6v+9-44=(v-3)^2-44`

so in order to solve it we can solve

`(v-3)^2=44`

By setting `x=v-3`, we can solve `x^2=44 Rightarrow x=2pm sqrt(11)`, hence `v-3=sqrt(11) Rightarrow v=3+sqrt 11`

Answer 2

`v=2sqrt(11)+3`

Explanation:

Completing the square is a process where one would manipulate the polynomial such that it could be factored into the form (variable +/- number) squared.

The trick to finding the third term in the polynomial is `(b/2)^2`. Thus, for this equation, we want the left side to be `v^2-6v+9`.

With that in mind, we add 9 to both sides, and simplify.
`v^2-6v+9=44`
`(v-3)^2=44`

Now we can take the square root of both sides, and solve for v.
`v-3=sqrt(44)`
`v=sqrt(44)+3`
`v=2sqrt(11)+3`