Question

What term must be added to 4x^2 - 12xk to make it a perfect square root?

Desperately Need help ! Can somebody show me the step? This is killing me. Thanks !!

Answer 1

if you visualize 4x^2 - 12xk + something to be perfect square
then it equals
(2x+something)^2

Answer 2

I would start by factoring out the 4 to get 4(x^2 - 3x_____). From there, we need to take the x^1 coefficient and "half it and square it"
So if you take half of 3 then square that result, what do you get?

Answer 3

I still don't get it :(.

Answer 4

Completing the Square 101
Imagine you have something of the form
\[\Large z^2 + \color{purple}pz = k\]
And for some reason or another, you need to express it in terms of a perfect square trinomial.

The first thing to watch out for is to make sure that the square bit... (this part)\[\Large \color{red}{z^2 }+ \color{purple}pz = k\] is unadorned... meaning, it has

no number next to it... if it does, well, remedy it by dividing the entire expression by that number...

So now, pay attention to the coefficient of the 'unsquared' variable, in this case, \(\large \color{purple}p\). What you need to do is halve it, resulting in \(\Large

\frac{\color{purple}p}{2}\) and then square it, yielding \(\Large \frac{\color{purple}p^2}{4}\).

Next, you add this to both sides of your equation, like so..
\[\Large z^2 + \color{purple}pz+\frac{\color{purple}p^2}4 = k+\frac{\color{purple}p^2}{4}\]

You'll notice that the left side is now a perfect square, in particular, the square of \(\Large z+\frac{\color{purple}p}2\)...
\[\Large \left(z+\frac{\color{purple}p}2\right)^2=k+\frac{\color{purple}p^2}{4}\]

And that's it!
You have successfully 'completed the square'
:)

[[EXAMPLE]]
[USING COMPLETING THE SQUARE TO SOLVE A SIMPLE QUADRATIC EQUATION]
Say we have \[\Large 4x^2 -8x=60\]First of all, you'll notice that the coefficient of the square \(\large x^2\) is not 1, so we had best get rid of that coefficient by dividing the entire expression by it (-4). And we get
\[\Large x^2 \color{purple}{-2}x=15\]
Now, the coefficient of the unsquared x is -2. Half of that is -1, and the square of -1 is 1.

We therefore add 1 to both sides of the equation, like so
\[\Large x^2-2x \color{blue}{+1}=15\color{blue}{+1} \]

Notice now that the left-hand side of the equation is now a perfect square trinomial, specifically, it is the square of \(\large x-1\).
Thus, the equation is as follows:
\[\Large (x-1)^2 = 16\]
now, extracting the square roots of both sides of the equation, (and be careful, you must consider the positive and negative square roots of the constant 16 in equations like this)
\[\Large x-1=\pm 4\]
\[\Large x = 1\pm 4\]Which means
x could either be
1+4 =5
or
1-4 = -3

Indeed, plugging either of these values in checks out for our original equation
\[\Large 4x^2 -8x = 60 \]

And that... is the basic principle of completing the square
^_^

Answer 5

Well with any quadratic ax^2 + bx + c. If a is 1, then you can take half of b and square it in order to find out the number that makes it a perfect square.
And holy hell @terenzreignz

Answer 6

No, @Psymon
It's a pre-made tutorial I make so that I don't have to type those long stuff all over again :)

Answer 7

I just browse through them and pick up whichever is relevant :3

Answer 8

I have one of those but for derivatives, haha.

Answer 9

May I have it? XD
I'll trade it for the quadratic formula one :3

Answer 10

Haha, alright. There is one bad thing about mine and it is that I dont have much about trig in it. I need to add that x_x But Ill send you the files xD

Answer 11

I was just kidding :)
And lol, files? I just have some text ready for copying and pasting into OS
and it just pops up like that ^holy hell tutorial XD

Answer 12

Yeah, mine were things I typed up in paint so I could add drawings and all. I should type my next tutorial stuff in the equation editor and all, lol

Answer 13

Sorry @huiyingtanyayap need a step-by-step demo? :)

Answer 14

Let me try it ! Thanks guys ! ;)

Answer 15

Can you show me the step ? I am using completing the square but I am stuck....

Answer 16

Did you factor out the 4?

Answer 17

Yes but still don't get it :(

Answer 18

So what you have is 4(x^2 - 3x)
Now you need to like I said "half it and square it" So we need to divide 3 in half, which gives us (3/2) Can you tell me what (3/2)^2 is?

Answer 19

9/4

Answer 20

Right :p So now we have 4(x^2 -3x + 9/4). Now just multiply the 4 back in

Answer 21

So the answer is 9 ?

Answer 22

Yep :3

Answer 23

Oh thanks!!,,

Answer 24

Mhm :3