Algebra Question

douglas14

Hi Im a first time poster, just wondering if you could help me with a few questions I was stuck with. I am a leaving cert student and I just couldnt solve these questions. Ive read the charter and know you cant get people to do homework for you, these are just a few things i couldnt get right while studying and would appreciate some help.

f(x) = x^2+px+p for all xeR, peZ

(i) find the range of values of p for which f(x) =0
(ii)the roots of f(x)=0 are o and (1+k)o
show that p-4 = k^2/1+k

Also this question

(i) a/b =c/d
show that (a-c)/(a+c)=(b-d)/(b+d)

Tried cross multiplication and came close, couldnt fnish this tho.

(ii)

m>n>0 where m, n eQ

show that n<(m+n)/2<m

This last question looks very easy and is easy to figure out in your head but i couldnt get full marks because of how i wrote it out.

Hope you can help thanks in advance.

MathsManiac

douglas14 wrote: »

f(x) = x^2+px+p for all xeR, peZ

(i) find the range of values of p for which f(x) =0
(ii)the roots of f(x)=0 are o and (1+k)o
show that p-4 = k^2/1+k

I think you haven't reproduced this question correctly. Can you check it?

douglas14 wrote: »

(i) a/b =c/d
show that (a-c)/(a+c)=(b-d)/(b+d)

Tried cross multiplication and came close, couldnt fnish this tho.

Show us what you've got so far.

douglas14 wrote: »

(ii)

m>n>0 where m, n eQ

show that n<(m+n)/2<m

Try this: start with the given fact m>n>0
Add m to everything and think about what you might do next.

douglas14

Thanks for the reply MathsManiac. for the first question you're right it should read
(i) find the range of values of p for which f(x) = 0 has real roots
(ii)the roots of f(x)=0 are o and (1+k)o (Im using o instead of alpha)
show that p-4 = k^2/1+k
Second question:
Actually after much thinking, I might have this right. well i deduced that for example c/d=b/d therefor b-c=0 and similarly a-d=0 so b-c=a-d so a+c=b+d. repeat similarly for top line. Hope this is right, know my explanation isnt great but let me know if its right if you can follow it.
3rd question:
tried to follow your advice and got 2m>m+n>m so m>(m+n)/2>m. Is this a sufficent way to solve the equation since m>n? As I said I think its very easy to understand why it is true from thinking about it but is this the best way to write it? Also is there any other way of doing it other than adding m to each part because that kind of method is unusual among leaving cert answers.

MathsManiac

For the first part of the first one, you need to use the fact that a quadratic equation has real roots when b^2 - 4ac = 0, (where the equation is ax^2 + bx + c = 0).

For the second part, you need to use the fact that the sum of the two roots is -b/a and/or the fact that the product of the roots is c/a.

You've gone a bit wrong on the next one. Try to keep the logic in mind while you're doing the simplifying: you have one statement that you are given as true, and another that you are trying to prove.

You can use cross-multiplication to get rid of the fractions, but you still need to think "This is what I know; and this is what I'm trying to show."

The thing you know is that ad = bc (by cross-multiplication).

Now try to simplify the thing that you're trying to prove. In a few steps, you should eventually be able show that it reduces to the thing you were given.

The third one has a little bit more to it than I thought at first, but note that you're half-way there. You've been asked to prove two things: that (m+n)/2 is smaller than m, and that it's bigger than n. You have proved one of these. Can you think of a similar argument that would allow you to prove the other?

Fremen

MathsManiac wrote: »

You've gone a bit wrong on the next one. Try to keep the logic in mind while you're doing the simplifying: you have one statement that you are given as true, and another that you are trying to prove.

You can use cross-multiplication to get rid of the fractions, but you still need to think "This is what I know; and this is what I'm trying to show."

The thing you know is that ad = bc (by cross-multiplication).

Now try to simplify the thing that you're trying to prove. In a few steps, you should eventually be able show that it reduces to the thing you were given.

Whole multiplying everything out is probably the right approach in this case, it's worth mentioning that in leaving cert maths in general, that approach is almost never the right one.

If something looks like it will take more than two lines to expand out, it's always worth taking a step back and looking for other ways to approach it first. Can't think of a good example right now, but I'll post one when I do.

douglas14

thanks for the help thus far. Just struggling with part (ii) question
p-4=k^2/(1+k). i got that -p=2o+ko and p=o^2+ko^2
k+1=p/o^2. i got k=(-p-2o)/o. k^2= (p^+4po+4o^)/o^2.
Anyway my answer didnt come close. Tried doing it a few different ways but not getting the right answer.

Have I made a mistake or should I approach it differently?

MathsManiac

Note that there's no alpha in the thing you're trying to prove, so you need to try to eliminate alpha.

There are a few ways to do this. Here's one:

Use you first equation to write alpha in terms of p and k.

Then substitute this into the second equation, so that you get an equation with "p"s and "k"s only.

Isolate p, and then write an expression for p-4, and then see if you can get it into the required form.

douglas14

Tried that way. had o=-p/2+k and substituted it in to this formula: p=0^2(1+k). ended up with p^2/k^2+4k+4=p/1+k. Cross multiplied that but couldnt finish it. seems close though. maybe I made a mistake along the way.

MathsManiac

You're pretty close.

Assuming p isn't 0, you can divide across by p.

Then mult across by the denominator on the LHS to get p on its own.

Then subtract 4 from both sides to give p-4, and you should be able to finish out.