Converting quadratic to perfect square

[forestearth]

Convert the quadratic equation y= ax^2+bx+c into a form involving a perfect square.
what do they mean by a perfect square???? i have the answer but dont understand what they are doing.
the answer says: assuming a=1, take one-half the coefficient of x (b/2), square it (b^2/4), and add and subtract the latter term so as not to change the value of the original function, then rearrange and factor to create a perfect square.
answer:
y = (x+b/2)^2 + (c-b^2/4)
how is this a perfect square?????

Please advise!

Thanks!

Al

 

[galactus]

This technique is called 'completing the square'.

Let's take a particular quadratic.

\(\displaystyle x^{2}+6x+5=0\)

\(\displaystyle x^{2}+6x=-5\)

Half of 6 is 3, 3 squared is 9. Add 9 to both sides:

\(\displaystyle x^{2}+6x+9=4\)

The left side is now a perfect square trinomial. That is what they mean.

Continuing:

\(\displaystyle (x+3)^{2}-4=0\)

They are asking you to do it in general. Assuming a=1:

\(\displaystyle x^{2}+bx+c=0\)

Half of b is b/2, square that and we get \(\displaystyle \frac{b^{2}}{4}\)

Add it to both sides:

\(\displaystyle x^{2}+bx+\frac{b^{2}}{4}=-c+\frac{b^{2}}{4}\)

See?. The left side is now a perfect square and can be written:

\(\displaystyle (x+\frac{b}{2})^{2}\)

 

[forestearth]

Thanks!
So when they say "perfect square" it has nothing to do with the shape square? it is just a name for an equation called a perfect square trinomial?

 

[tkhunny]

Generally, the process of "completing the square" doesn't use the word "perfect".

 

[mmm4444bot]

… into a form involving a perfect square.

There may be a word missing, in this phrase.

Many times I have seen this written as, "… into a form involving a perfect square trinomial".

A trinomial is any polynomial with three terms.

The phrase "perfect square trinomial" refers only to those trinomials that can be factored as the square of some expression, like (x+b/2)^2, for example.

By the way, this does involve a square, if you interpret it geometrically.

Cheers ~ Mark

 

[forestearth]

So trinomials that can be factored as
the square of some expression always involve a geometric square?

 

[mmm4444bot]

So trinomials that can be factored as the square of some expression always involve a geometric square?

Close, and yes, I meant a geometric square.

I did not really spell-out what I meant by my "square" statement because it's not really important. Just silly trivia.

I mean that trinomials that can be factored as the square of some expression can always be interpreted as having something to do with the sides of a square. In other words, you can alway contrive some scenario, if you're working outside the context of a word problem, that involves a geometric square, as (expression)^2 can always be interpreted as the area of a square with side length "expression".

The main points that I want to get across with my first post is that (1) I think they want you to get from the phrase "form involving a perfect square" that your answer will involve a factorization that looks like (x - something)^2, and (2) if you expand such an expression the resulting polynomial is called a "perfect square trinomial" because the trinomial factors into two identical factors , so that's where the word perfect comes from. (In other words, they left off the word trinomial.)

Do you get the symbolic square-completion posted by galactus? That's the point of the exercise; to see if you can work through the algorithm of "completing the square" using symbolic coefficients versus Real numbers.

Cheers ~ Mark

 

[forestearth]

Thanks!You guys are awesome! so its basically the idea that the area of a square = side^2.
I guess the point of the symbolic square completion is to demonstrate this alternative way of solving quadratic equations?
is there a certain type of problem where generally completing the square comes in handy?

this is a great site!


<<<<Do you get the symbolic square-completion posted by galactus? That's the point of the exercise; to see if you can work through the algorithm of "completing the square" using symbolic coefficients versus Real numbers.>>>>>>

 

[mmm4444bot]

so its basically the idea that the area of a square = side^2.

It could be. I mean, you'll probably wind up with some word problem in the future that involves the area of a square thing where you model that area with a quadratic polynomial. But there are also many "quadratic" word problems that do not involve the area of a square. So, not all problems are "squares", heh, heh.


I guess the point of the symbolic square completion is to demonstrate this alternative way of solving quadratic equations? Exactly.

is there a certain type of problem where generally completing the square comes in handy?

Oh yes. Many types!

If you are taking a bonafide course, then you are now simply learning how to use the tool of completing squares. Eventually, you will apply this algorithm many times to accomplish all sorts of things, especially if you continue in your math studies.

Oh. It's kinda funny that completing-the-square is often used when working with the algebraic equations of circles, but true.

Starting with form Ax^2 + By^2 + Cx + Dy = E, you will need to arrive at the following form.

r^2 = (x - h)^2 + (y - k)^2

That's the equation for a circle of radius r centered at the point (h, k).

Two quadratic polynomials up there.