Quadratic Equations and 13 year old boys...

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tomandlu
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Quadratic Equations and 13 year old boys...

Postby tomandlu » Wed Mar 14, 2012 11:42 am UTC

Hi,

My son's homework last night consisted of finding various products and sums or their factors - e.g. x+3.1 = 5.3 or x * 2.5 = -6.7

So far, so good... but the final problem gave the product and sum of x and y, and asked him to find x and y... which as far as I know is solvable, but is not simple. After some practice, I can now successfully find the factors via completing the square, but this seems very much in at the deep end...

Am I missing something here? Is there a simpler approach to solving xy = -13.8 and x+y = 4.3 than completing the square?

(I ended up with (x - 2.15)(x - 2.15) - 18.4225 = 0 -> x - 2.15 = 4.29)

I suppose, as an alternative, the teacher is expecting it to fox them, and just wanted them to understand it was a complex problem. Would solving this sort of thing be considered normal for a 13 year old or is there a simpler way?
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Re: Quadratic Equations and 13 year old boys...

Postby masher » Wed Mar 14, 2012 11:59 am UTC

xy = -13.8 and x+y = 4.3

y=-13.8/x

x + (-13.8/x) = 4.3

Solve for x.

Substitute in to solve for y.

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Re: Quadratic Equations and 13 year old boys...

Postby jaap » Wed Mar 14, 2012 12:22 pm UTC

masher wrote:xy = -13.8 and x+y = 4.3

y=-13.8/x

x + (-13.8/x) = 4.3

Solve for x.

Which still gives to the quadratic equation x^2 - 4.3 x -13.8 = 0, which you either solve by completing the square or by blindly applying the quadratic formula.
There is no way to work around the fact that it is quadratic, so no way to avoid completing square / quadratic formula. (Well, apart from somehow guessing the solution but the numbers are a bit too awkward in this case.)

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Re: Quadratic Equations and 13 year old boys...

Postby tomandlu » Wed Mar 14, 2012 12:25 pm UTC

masher wrote:xy = -13.8 and x+y = 4.3

y=-13.8/x

x + (-13.8/x) = 4.3

Solve for x.

Substitute in to solve for y.


How do I solve that for x? I'm probably missing something here, but that doesn't look like it has an easy solution either... I should re-emphasise, the problem isn't finding a solution, the problem is finding a solution that a 13 year old would be expected to work out...

Edit to add: Thanks, Jaap. I thought I was being thick...
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Re: Quadratic Equations and 13 year old boys...

Postby Meteoric » Wed Mar 14, 2012 12:36 pm UTC

In simpler cases (when the numbers involved are sufficiently small integers), you can attempt to solve them basically by exhaustion. For example, given xy = 100 and x+y=29, you can look at the factors of 100 and see which pair adds to 29, or you can look at pairs that add to 29 (1+28, 2+27, 3+26, etc) and see which multiply to 100. But that doesn't work so well once you're dealing with things other than just integers, and there's no guarantee that the answer will be an integer even if the question only contained integers, so unless you know the problem will be "nice" it isn't a good method.
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Re: Quadratic Equations and 13 year old boys...

Postby tomandlu » Wed Mar 14, 2012 12:41 pm UTC

Meteoric wrote:In simpler cases (when the numbers involved are sufficiently small integers), you can attempt to solve them basically by exhaustion. For example, given xy = 100 and x+y=29, you can look at the factors of 100 and see which pair adds to 29, or you can look at pairs that add to 29 (1+28, 2+27, 3+26, etc) and see which multiply to 100. But that doesn't work so well once you're dealing with things other than just integers.


Indeed - the solutions to this were 6.44 and -2.14.

I'm curious to see what he says when he comes back this afternoon. He's in the top-stream for maths, but afaik this is still way out of his league and experience...
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Re: Quadratic Equations and 13 year old boys...

Postby tckthomas » Wed Mar 14, 2012 12:46 pm UTC

aren't quadratic equations taught at age 13?

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Re: Quadratic Equations and 13 year old boys...

Postby Meteoric » Wed Mar 14, 2012 12:49 pm UTC

At age 13 he's in... what, 7th, 8th grade? I think that's around when the more advanced students do Algebra, but I cannot pretend to know the curriculum everywhere in the world, and I obviously don't know your son specifically. If he's in algebra, he might have done completing the square or the quadratic formula, but if so he presumably would have at least recognized material from earlier in the year. Do tell us what the teacher says :)
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Re: Quadratic Equations and 13 year old boys...

Postby tomandlu » Wed Mar 14, 2012 3:20 pm UTC

He's back...

The teacher didn't expect anyone to get it (although one child found an approximate answer through trial and error).

Teacher explained that it was solvable, and mentioned quadratic equations and completing the square, with an example (which they were not expected absorb).

They cover this in about 2 years time... phew...
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Re: Quadratic Equations and 13 year old boys...

Postby eSOANEM » Wed Mar 14, 2012 4:29 pm UTC

I know it's a bit late now, but, being an 18-year-old UK student, I remember roughly what I was expected to know at 13 and I think that my approach (if I continued beyond "I don't know how to do this") would probably have been to try plotting the two graphs and graphically finding an approximation to the solutions.
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Re: Quadratic Equations and 13 year old boys...

Postby tomandlu » Wed Mar 14, 2012 5:30 pm UTC

eSOANEM wrote:I know it's a bit late now, but, being an 18-year-old UK student, I remember roughly what I was expected to know at 13 and I think that my approach (if I continued beyond "I don't know how to do this") would probably have been to try plotting the two graphs and graphically finding an approximation to the solutions.


Yes, we did contemplate this (graphing and seeing where they crossed). The main problem was that the conversation last night went something like this:

Son: Dad, can you help me with some maths homework?
Dad: Sure - just let me finish this large glass of wine...

After that, we got as far as:

sx - (x*x) - p = 0

I knew there was a way to solve that, but didn't track down the solving the square solution until I got up this morning... got to admit, I enjoyed myself. Haven't done any proper algebra for a long time...
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Re: Quadratic Equations and 13 year old boys...

Postby Qaanol » Wed Mar 14, 2012 6:22 pm UTC

Yep, completing the square is pretty nice. There’s an easy way to picture what’s going on, that shows how it got the name “completing the square”:

Spoiler:
Starting from a general quadratic a·x2 + b·x + c, divide through by ‘a’ to get x2 + (b/a)x + (c/a). Then draw a picture of what that means in terms of areas of rectangles (the “c/a” block has height 1, sorry for the poor handwriting):

Image
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Re: Quadratic Equations and 13 year old boys...

Postby tomandlu » Wed Mar 14, 2012 6:35 pm UTC

Qaanol wrote:Yep, completing the square is pretty nice. There’s an easy way to picture what’s going on, that shows how it got the name “completing the square”:

Spoiler:
Starting from a general quadratic a·x2 + b·x + c, divide through by ‘a’ to get x2 + (b/a)x + (c/a). Then draw a picture of what that means in terms of areas of rectangles (the “c/a” block has height 1, sorry for the poor handwriting):

Image


Now I know I'm being thick... what does the c/a box do? It just seems to sit there like a big fat box. I can't see where it becomes useful... It must be the extra constant, but I'm lost... (hard to read lettering doesn't help an old(ish) man...)
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Re: Quadratic Equations and 13 year old boys...

Postby gorcee » Wed Mar 14, 2012 7:10 pm UTC

You have ax^2 + bx + c, right? Now, divide through by a to get x^2 + b/a * x + c/a.

Each one of these terms is a box in the start of the drawing. There's a box for x^2, a box for b/a * x, and a box for c/a

What we do is we take that b/a * x box, chop it in half, and slap it onto the x^2 box. But, this doesn't make a nice square -- there's a notch cut out. So, to fill in the diagram to make it a perfect square, we have to borrow an extra bit from the rest of the paper, and then account for it later -- kind of like transferring 10 quid from your savings to your checking account to make up for a parking ticket you didn't expect. This "borrowed" bit is balanced out by adding a negative red square.

During this whole process, the c/a box is pretty lonely. And that's OK! We don't need to do anything to it; we can just let it be.

The important part of this was taking the x^2 term and transforming it into something that looks like (x+k)^2, where k is determined through our hacking apart the b/a*x box (and borrowing a little from the banker).

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Re: Quadratic Equations and 13 year old boys...

Postby Qaanol » Wed Mar 14, 2012 9:38 pm UTC

tomandlu wrote:
Qaanol wrote:Yep, completing the square is pretty nice. There’s an easy way to picture what’s going on, that shows how it got the name “completing the square”:

Spoiler:
Starting from a general quadratic a·x2 + b·x + c, divide through by ‘a’ to get x2 + (b/a)x + (c/a). Then draw a picture of what that means in terms of areas of rectangles (the “c/a” block has height 1, sorry for the poor handwriting):

Image


Now I know I'm being thick... what does the c/a box do? It just seems to sit there like a big fat box. I can't see where it becomes useful... It must be the extra constant, but I'm lost... (hard to read lettering doesn't help an old(ish) man...)

If your quadratic equals zero, as it does in most practical problems, then you have x2 + (b/a)x + c/a = 0. After completing the square we have (x + b/2a)2 + c/a - (b/2a)2 = 0. We can expand out (b/2a)2 = b2/4a2. Then we can move the constants to the other side and get:

(x + b/2a)2 = b2/4a2 - c/a

The constants can be put into one fraction with a common denominator as:

(x + b/2a)2 = b2/4a2 - 4ac/4a2
(x + b/2a)2 = (b2 - 4ac)/4a2

The right-hand side is just a number, and the left-hand side is a perfect square. Getting to this point was the purpose of completing the square. So now we can take the square root of both sides:

x + b/2a = ±√(b2 - 4ac)/2a

Then bringing the last constant over to the right side gives:

x = -b/2a ± √(b2 - 4ac)/2a

The stuff on the right already has a common denominator, so we can write it as a single fraction:

x = (-b ± √(b2 - 4ac))/2a
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