Will somebody show me how to turn this parabola equation into vertex form?

Question

The equation is written:

y = 3(x+2)(x+3)

Here is what I did:

First I factored the equations: x^2 + 3x + 2x + 6
Then I added in the 3 to get: 3x^2 + 15x + 18
Then I got this: 3(x^2 + 5x) + 18
Then I did this: 3(x^2 + 5x + 6.25) + 18

Now I need to factor this part >> (x^2 + 5x + 6.25) << HELP!

Answer 1

You were doing fine up to here:

3(x^2 + 5x) + 18

In this step,

3(x^2 + 5x + 6.25) + 18

you found the correct number to "complete the square", but having added it in, you must subtract it out again; otherwise, you've changed the problem. So what I would suggest is something like this:

3(x^2 + 5x + 6.25 - 6.25) + 18

which is like adding zero, and then grouping like this:

3((x^2 + 5x + 6.25) - 6.25) + 18

Now, to x^2 + 5x + 6.25. You found the 6.25 by taking half of the coefficient of the x term, right? Well, don't forget the 2.5, because it's exactly what you need to do the factoring:

x^2 + 5x + 6.25 = (x + 2.5)^2

so you get

y = 3((x + 2.5)^2 - 6.25) + 18

Now multiply the 3 by the two terms in the parentheses, and combine the constants:

y = 3(x + 2.5)^2 - 18.75 + 18 = 3(x + 2.5)^2 - 0.75

and now it's in vertex form [technically, we should write it as 3(x - (-2.5))^2 - 0.75 ]

Then we can read off that the vertex is at (-2.5, 0.75)