completing the square

There are different ways to think about 'completing the square' but I recommend this method when there are fractions and it's non calculator:

E.g. $x^2 - 3x + 5$

Start by halving $-3$ to get $-\frac{3}{2}$ as you say

So you get $\left(x-\frac{3}{2}\right)^2$

Then you need to subtract the square of and then add on 5 (let me know if you can't see where I'm getting these numbers from).

So you end up with :

$\displaystyle \left(x-\frac{3}{2}\right)^2 - \left(-\frac{3}{2}\right)^2 + 5$

$\displaystyle =\left(x-\frac{3}{2}\right)^2 - \frac{9}{4} + 5$


Now change the $5$ so it has the same denominator as $\frac{9}{4}$

$\displaystyle =\left(x-\frac{3}{2}\right)^2 - \frac{9}{4} + \frac{20}{4}$

$\displaystyle =\left(x-\frac{3}{2}\right)^2+\frac{11}{4}$


Now see if you can follow this process for a different example. Post your working if you get stuck.

as notnek said, try do an example so we can see where you get stuck.

try complete the square on $\displaystyle x^2 - 3x + 5$

i think i got it

gimme an example n how do u write in that format cox idk it seems easy to follow that way :/

First have a go at $x^2-5x+12$ which is similar to the last example. Please post your answer or show your working if you get stuck.

I typed it using LaTeX but you don't need to use that - just make sure your working is clear when you post so use brackets

e.g. (x-(3/2))^2 - (9/4) + 5

ok

(x - 5/2)^2 - 25/4 + 12

(x - 5/2)^2 - 25/4 + 48/4

(x - 5/2)^2 + 23/4

is that right :/

x^2 - 3x + 5 in the form (x - p)^2 + q
can anyone plx explain how do i do completing the square because i dont know if there is a method to do them
i suck at maths lol

Correct

Now try $x^2+ 9x - 15$ and then $x^2-13x-4.5$. Change the decimal to a fraction for the second one.

If you can do them then I think you're done. The topic does get harder when you have something like $2x^2+7x -3$ but that might be for another day!

A lot of replies!

We've three tips for completing the square. We are going to make your example x^2 - 3x - 5 as it is a simpler case.

The first tip is to divide through by a whenever a is greater than 1 in ax^2 + bx + c. Not a problem in your example since a = 1 so something to remember for the future.

Working with the example, the second tip is to write x^2 - 3x = 5 and IMMEDIATELY add 9/4 to both sides ie (b/2a)^2. So you have x^2 - 3x + 9/4 = 7.25

Then 'complete the square' : (x - 3/2)^2 = 7.25 (on the LHS - 9/4 + 9/4 cancels out). Then you can square-root both sides:

x - 3/2 = SQRT 7.25

And so x = 3/2 +/- SQRT 7.25

Our third tip is always to make assure you have plus AND minus the square root since it is easy to forget to do this and lose marks.

Lastly, examiners like to ask you to write down the value of x that minimises or maximises the function of x. This is to test that you know one of the useful features of writing ax^2 + bx + c in the form p(x + q)^2 + r.

The very useful thing to know is that if a and p are greater than zero, the function is U-shaped and so has a minimum. Conversely if a and p are less than zero, the function is an upside-down U and so has a maximum. The value of x that gives you the minima or maxima is the one that makes the brackets bit equal zero. So in the example, x = 3/2 gives a minima.

...
Working with the example, the second tip is to write x^2 - 3x = 5 and IMMEDIATELY add 9/4 to both sides ie (b/2a)^2. So you have x^2 - 3x + 9/4 = 7.25

Then 'complete the square' : (x - 3/2)^2 = 7.25 (on the LHS - 9/4 + 9/4 cancels out). Then you can square-root both sides:

x - 3/2 = SQRT 7.25

And so x = 3/2 +/- SQRT 7.25
...

ok so the first one...

(x - 9/2)^2 - 81/4 - 15

(x - 9/2)^2 - 81/4 - 60/4

(x - 9/2)^2 - 141/4 --------> (i feel like its wrong because i end up with a huge number :/ )

2nd one...
(x - 13/2)^2 - 169/4 - 9/2 (45/10 simplifies to 9/2)

(x - 13/2)^2 - 169/4 - 16/4

(x - 13/2)^2 - 175/4

i think both r wrong O_O




thnx for the tips
they r very helpful +_+

You're very close in both but they both need slight corrections.

For the first one notice that the quadratic I gave you contained $+9x$ instead of $-9x$ so you need to change one of the signs in your answer.

For the second one you said 9/2 is the same as 16/4. Check that again.

oh yeah its +9 i always miss the signs so its...
(x + 9/2)^2 - 141/4 ?

i did update my 2nd one again but maybe u didnt got that yet :/
i'll post it here again

2nd one...
(x - 13/2)^2 - 169/4 - 9/2 (45/10 simplifies to 9/2)

(x - 13/2)^2 - 169/4 - 36/4

(x - 13/2)^2 - 205/4

The first one is now correct but for the second one 9/2 is not equal to 36/4

I have to go now but hopefully someone else will continue to help tonight if you need it or want to try harder examples. @RDKGames is always helpful if he's free.

oml i am so stressing out why am i not getting it right arghhh

ok so

the qs is x^2 -13x - 4.5

solution:

(x - 13/2)^2 - 169/4 - 9/2

x - 13/2)^2 - _________ (i am not getting that bit right i think.... do i times 9/2 by 2 to get 18/4???)

@RDKGames plx be easy on me

Yeah that's right $\frac{9}{2}=\frac{18}{4}$