Completing the square? "solve 2m-m^2 by completing the square."

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kmce

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I am trying to solve 2m-m2 by completing the square.

I know the answer is meant to be 1 – (1 – m)2 but i cant figure out how it got there. The closest ive gotten is -(-1+m)2 -1

Can anyone lend me some help to how to solve this the correct way
 
Well, unfortunately, we can't troubleshoot work we can't see. You've told us the end result you got, but not any of the steps you took to get there. For instance, what did you get when you tried following the standard steps for completing the square? Please share with us all of the work you've done on this problem, even the parts you know for sure are wrong. Thank you.
 
kmce said:
I am trying to solve 2m-m2 by completing the square.

I know the answer is meant to be 1 – (1 – m)2 but i cant figure out how it got there. The closest ive gotten is -(-1+m)2 -1

Can anyone lend me some help to how to solve this the correct way
Click to expand...
You have given an expression not an equation. You can't solve an expression.
 
That was all the information i was given. The question says

Complete the square for the quadratic and hence, state its maximum or minimum valueand the point at which this value is attained – sketch the quadratic roughly: 2m – m2

So i started the same way as if it were x2+ 6x + 10

I took out -1 as the common factor so got

-1(m2+2m)
then i started completing the square as normal

-1 (m+1)2

Im not completely sure if i am meant to square the k value as you would normally do since i dont have a p value. But even if i was i would end up with (m+1)2
 
mmm4444bot said:
You have not defined the meaning of any symbols k or p, so I'm not sure what you're talking about with those symbols.

Correct your initial factorization, and try again.

If you still need help, please tell us the value of the coefficients A,B,C in the standard form of the given quadratic polynomial:

Am^2 + Bm + C :cool:
Click to expand...

The equation we work off of is

y = a(x + k)2 + p
 
kmce said:
That was all the information i was given. The question says

Complete the square for the quadratic and hence, state its maximum or minimum valueand the point at which this value is attained – sketch the quadratic roughly: 2m – m2

So i started the same way as if it were x2+ 6x + 10

I took out -1 as the common factor Good first step, but ...

-1(m2+2m) ... your factorization is not correct.

You factored -2m-m^2 instead of 2m-m^2

then i started completing the square as normal

-1 (m+1)2 This result does not match the given quadratic, either.

If we expand your expression (i.e., multiply out everything), we get

-m^2 - 2m - 1 instead of -m^2 + 2m
Im not completely sure if i am meant to square the k value as you would normally do since i dont have a p value. But even if i was i would end up with (m+1)2
Click to expand...
You have not defined the meaning of any symbols k or p, so I'm not sure what you're talking about with those symbols.

Correct your initial factorization, and try again.

If you still need help, please tell us the value of the coefficients A,B,C in the standard form of the given quadratic polynomial:

Am^2 + Bm + C :cool:
 
kmce said:
The equation we work off of is

y = a(x + k)2 + p
Click to expand...

This is called "vertex form". Are you sure they gave this form to you with a plus sign (highlighted in red) instead of a minus sign?

Anyway, this form is the end result, not the starting point, in your exercise.

In other words, we complete the square on the right-hand side of

y = -m^2 + 2m

to arrive at the form

y = a(m - k)^2 + p

With a minus sign instead of a plus sign, we can read off the vertex coordinates as (k,p).

Did you correct your initial factorization?
 
kmce said:
That was all the information i was given. The question says

Complete the square for the quadratic and hence, state its maximum or minimum valueand the point at which this value is attained – sketch the quadratic roughly: 2m – m2

So i started the same way as if it were x2+ 6x + 10

I took out -1 as the common factor so got

-1(m2+2m)
then i started completing the square as normal

-1 (m+1)2

Im not completely sure if i am meant to square the k value as you would normally do since i dont have a p value. But even if i was i would end up with (m+1)2
Click to expand...

2m – m2 = -(m2 - 2m +1) + 1 = -(m + 1)2 + 1
 
mmm4444bot said:
This is called "vertex form". Are you sure they gave this form to you with a plus sign (highlighted in red) instead of a minus sign?

Anyway, this form is the end result, not the starting point, in your exercise.

In other words, we complete the square on the right-hand side of

y = -m^2 + 2m

to arrive at the form

y = a(m - k)^2 + p

With a minus sign instead of a plus sign, we can read off the vertex coordinates as (k,p).

Did you correct your initial factorization?
Click to expand...

Yup, it was a plus sign. The slide i am working off says the following

A quadratic function y = ax2 + bx + c can be written in the form y = a(x + k)2 + p, for suitable constants k and p. This is called completing the square.

I havent went back over and tried it yet, I was working on some triq, but will do so now :)
 
kmce said:
Can i ask where the +1 came from (highlighted in red).
Click to expand...

That's what was added to "complete" the square.

If we start with the form x^2 + Bx, then we add (B/2)^2 to complete the square.

We do this because x^2 + Bx + (B/2)^2 factors as (x + B/2)^2


Your exercise uses the variable m instead of x.

-m^2 + 2m

This is not in the form x^2 + Bx because of the leading negative sign. So we factor out -1 to start.

-(m^2 - 2m)

Now we have the form x^2 + Bx inside the parentheses, with B=-2. We add (B/2)^2 inside the parentheses.

(B/2)^2 is (-2/2)^2 which is (-1)^2 = 1

-(m^2 - 2m + 1)

Now comes an important part: by adding 1, we have changed the given quadratic. In order to cancel this change (and return to a form equivalent to what we were originally given), we need to add 1 outside.

You might at first think we would need to subtract 1, but remember the -1 that we factored out at first.

-(m^2 - 2m + 1)

The negative sign out front changes the sign of every term inside the parentheses. So the 1 we added inside actually represents a -1. This is why we add 1 outside.

-(m^2 - 2m + 1) + 1

Because we completed the square inside the parentheses, that part now factors as (m+B/2)^2. And B/2 is -1, as we calculated above.

-(m - 1)^2 + 1

Let's verify that this vertex form is equivalent to the original quadratic given, by multiplying everything and simplifying.

-(m^2 - 2m + 1) + 1

-m^2 + 2m - 1 + 1

-m^2 + 2m

It checks. :)

This page explains the steps and shows graphically why adding (B/2)^2 to the form x^2 +Bx is known as "completing" the square:

https://www.mathsisfun.com/algebra/completing-square.html
 
Last edited:
Nice explanation Otis. Very detailed and thoughtful!

Here is a get-away-from-the-corner-free card for you.
 
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