Question

How to do this specific problem by completing the square, PLEASE help:
Write the quadratic function in vertex form.

y = x^2 + 8x + 18

Answer 1

@welshfella

Answer 2

what are the exact simplified steps to solving a completing the square?

Answer 3

first convert the x^2 + 8x to the form ( x + a)^2 - b

first divide the coefficient of x by 2 ( that is 8 /2) = 4
then you subtract (a/2) ^ 2

Answer 4

so we have

x^2 + 8x = (x + 4)^2 - 4^2

Answer 5

if you expand you'll see that this is a true identity
(x + 4^2 - 16 = x^2 + 8x + 16 - 16 = x^2 + 8x

Answer 6

oh ok- will these "same" steps work for solving the other completing the square problems as well?

Answer 7

Yes except if the coefficient of x s-s greater than 1
for example

4x^2 + 14x - you would first take out the 4

= 4 (x^2 + (14/4) x) and complete the square on whats in the parentheses

Answer 8

oh ok, and then so for this problem so far we have x^2+8x= (x+4)^2-4^2, so what do i do next?

Answer 9

we have
y = (x + 4)^2 - 16 + 18

y = (x+4)^2 + 2

Answer 10

how'd you get that

Answer 11

you need to remember to divide the coefficient of x by 2 then subtract the square of this outside the brackets

Answer 12

well it was originally y = x^2 + 8x + 18

Answer 13

oh ok got it

Answer 14

You could check this result by multiplying out the square of (x+4) and adding 2. Does that take you back to the original y = x^2 + 8x + 18?

Answer 15

if you can remember identities

x^2 + ax = (x + (a/2)) ^2 - (a/2)^2

Answer 16

is that the identity of completing the square

Answer 17

yes

Answer 18

in our case a = 8 and of course a/2 = 4

Answer 19

oh ok, and are there any shortcuts to this?

Answer 20

hmm. none that i know of!

Answer 21

y = (x + 4)^2 + 2 is vertex form

the graph would be a parabola with vertex (4 , 2)

Answer 22

ok!

Answer 23

While the "formula" discussed above is completely correct, it might be easier to remember the following steps:

y = x^2 + 8x + 18

1. Take half of the coefficient of x (which here is 8);
2. Square that result (which here produces 4^2 = 16)
3. Add that last result (16) to x^2 + 18x, and then subtract that same quantity:

y = x^2 + 8x + 16 - 16 + 18, which simplifies to y=(x+4)^2+2

That's it.

This last result has the form y=(x-h)^2+k, where (h,k) is the vertex.

Can you now identify h and k?

Write out the original equation, y = x^2 + 8x + 18, in this new format.

Answer 24

Oh yes I see the error


its x-h so the vertex will be at (-4,2)

Answer 25

That's right. @iwanttogotostanford : Any further questions?

Answer 26

oh ok, thanks both of you helped a lot!

Answer 27

yw

Answer 28

so @welshfella the answer that we concluded to is still right, correct?

Answer 29

y=(x+4)^2 + 2 is indeed correct. Make certain that you know how to obtain the coordinates of the vertex. What are they?

Answer 30

yes

y = (x + 4)^2 + 2 is the function in vertex form

The general form for the vertex is

y = (x - h)^2 + k


so comparing this with our function -h = 4 so h = -4 and k = 2

so the vertex (h , k) = (-4,2)

They didn't ask for the coordinates of the vertex I just added it as an extra.

Answer 31

its worth committing the formula for vertex to memory

In UK we used to use a and b for the coordinates but now they use h and k . ( I don't know why!!)

Answer 32

as practice can you complete the square of

x^2 - 10x ?

Answer 33

yes I will try x^2+25 @welshfella

Answer 34

thats as far as i got... i followed the steps and... yeah..

Answer 35

did i do it right?

Answer 36

Given:

x^2 - 10x

1. find half of the coefficient of the x term.
2. Square this result.
3. Add this square, and then subtract this square, to x^2-10x.
4. Rewrite the perfect square trinomial as the square of a binomial.

Refer to previous explanations; this is nothing new or different.
Please show your work.

Answer 37

gtg right now

Answer 38

@mathmale how do I do step 4 again? I did everything and got x^2-1x+25-25, but I don't know how to make that into a binomial

Answer 39

@mathmale

Answer 40

1. find half of the coefficient of the x term.
2. Square this result.
3. Add this square, and then subtract this square, to x^2-10x.
4. Rewrite the perfect square trinomial as the square of a binomial.

x^2 - 10x


1. half the coeff of the x term is half of -10, or -5.
2. The square of this result (-5) is +25.
3. Starting with x^2 - 10x, add 25, and then subtract 25: x^2 - 10x + 25 - 25
4. Rewrite x^2 - 10x + 25 as the square of a binomial