Equivalent Expressions: ""The area (in square meters), a, of a yard is given by...."

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Bronn

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Equivalent Expressions: ""The area (in square meters), a, of a yard is given by...."

Hi, first post, and probably not my last lol


Ive been using Kahn Academy to teach myself maths again from the begining. One of the frustrating things about that site is sometimes they start throwing questions at you that none of their videos seemed to have covered, atleast not fully. Leaving me to scratch my head a bit.

One such question is:
"The area (in square meters), a, of a yard is given by the expression:

a=x^4+18x^2+81

Where X is a positive integer and each dimension is greater than 1 meter. What are two (not necessarily unique) expressions that could represent the dimensions of the yard ?

In the answer box, write these two expressions as a product that equals the area (e.g. "(expression 1)(expression 2)")."

The answer and "working out" are:

step 1: x^4+18x^2+81 = (x^2 + 9)^2


Step 2: (X^2+9)(X^2+9)





I understand how to go from step 1 to step 2. But what I can't figure out is how they got step 1. Theres no method shown, and all the videos leading up to this had no similar examples, and I understood them all up to that point.

What I understand is, they want an equivelant expression that is the product of two expressions.
I see the the 81 was made into 9^2 and the x^4 became (x^2)^2? but I can't seem to figure where does the 18x^2 fit into this?

thanks for any help. Also more resources that cover this would be good if you know any.
I don't like to move on until i understand this fully
 
Last edited:
Alright, so just to make sure I'm understanding your question correctly, you were given this initial expression:

\(\displaystyle x^4+18x^2+81\)

And you're wondering what the process was to transform that into the following:

\(\displaystyle x^4+18x^2+81=(x^2+9)^2\)

So, this expression appears at first to be a bit intimidating, especially if you've not had much experience with factoring quartics (4th degree polynomials). However, in this case, it's made a lot simpler by the fact that there's neither an x3 term nor an x term. When faced with a polynomial like this, you can "shrink" it into a quadratic which you should be able to solve easily. Try using the substitution u=x2. Then rewrite the quartic in terms of u. Factor that, then back-substitute to end up in terms of x again.
 
yeah, you're right thats what Im confused on.


I feel the order of Kahn academy is all muddled up, because Ive been working thru introduction to algebra and its only covered basic things so far, like variables, and simple factoring. They haven't even defined polynomial, or qaudratic yet, then these questions pop up.
So basically most of what you've said has gone over my head.

Im going to have to find another resource to work thru....

thanks for the reply
 
Bronn said:
yeah, you're right thats what Im confused on.


I feel the order of Kahn academy is all muddled up, because Ive been working thru introduction to algebra and its only covered basic things so far, like variables, and simple factoring. They haven't even defined polynomial, or qaudratic yet, then these questions pop up.
So basically most of what you've said has gone over my head.

Im going to have to find another resource to work thru....

thanks for the reply
Click to expand...

The expression \(\displaystyle x^4 + 18x^2 +91\) is what I call a "hidden" quadratic. You will come across a lot of them.

Notice that \(\displaystyle x^4 + 18x^2 +81 = (x^2)^2 +18x^2 +81\)

Let \(\displaystyle x^2 = u\) and you get:

\(\displaystyle u^2 + 18u +81\) which I'm sure you can easily factorise into \(\displaystyle (u+9)(u+9)\) or \(\displaystyle (u+9)^2\).

Substituting \(\displaystyle u=x^2\)back you get \(\displaystyle (x^2+9)^2\).

Keep this in mind when you have an expression with three terms which you need to factorise. They sometimes reduce down to a simple quadratic.
 
Thanks for the reply.

However even with the 'u' i find myself stuck i the same spot. Its just not something I've covered in lessons yet. Im just going to have to find another resource because I can't follow the way Kahn is going.
Thanks for trying
 
Bronn said:
Thanks for the reply.

However even with the 'u' i find myself stuck i the same spot. Its just not something I've covered in lessons yet. Im just going to have to find another resource because I can't follow the way Kahn is going.
Click to expand...
Have you studied how to factor quadratics yet? Or how to do special factoring, like differences of squares or sums and differences of cubes? If not, then you may be interested in the following:

. . . . .Special Factoring

. . . . .Factoring Quadratics

Have fun! ;)
 
stapel said:
Have you studied how to factor quadratics yet? Or how to do special factoring, like differences of squares or sums and differences of cubes? If not, then you may be interested in the following:

. . . . .Special Factoring

. . . . .Factoring Quadratics

Have fun! ;)
Click to expand...

thanks for the resources.

No, only a tiny bit. not any with mixing additions/multiplications etc.
Thats whats frustrating me with Kahn academy atm. They sometimes throw in stuff they haven't covered yet in the practice sections. And I've been meticulously and systematically going thru every lesson in order. Ive noticed comments on videos of others complaining about similar things. Annoying!

Thanks again
 
Bronn said:
Thats whats frustrating me with Kahn academy atm. They sometimes throw in stuff they haven't covered yet in the practice sections.
Click to expand...
My impression is that Khan Academy is just a bunch of stand-alone lessons which have been thrown together in one place. There is no organizing principle, and any "ordering" is going to end up being confusing, oftentimes, especially if you're going there for learning, as opposed to a little refresher after class. :(
 
stapel said:
My impression is that Khan Academy is just a bunch of stand-alone lessons which have been thrown together in one place. There is no organizing principle, and any "ordering" is going to end up being confusing, oftentimes, especially if you're going there for learning, as opposed to a little refresher after class. :(
Click to expand...
Its fairly well organised. you can study thru k-12 order or thru subjects. Ive used it on and off for a while and its continuously being updated, so I think its come along way since the earlier days.
But every now and then theres a little mismatching going on which drives me crazy because i like sticking to the plan laid out.
 
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