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C1 Substitution Q

By substituting

t = x^{1/2}

, or otherwise, find the values of x for which:

4x+8 = 33x^{1/2}

Now, I'm stuck.

So;

4x+8=33t

Now what, as I have no idea what to do?
---------------

I mean, without substituting, I would have squared the terms and re-arranged to get;

4x^2-33x+8

and then solved some how (factorising/complete square/quad formula) to get x, but it specifically mentions substitution?

+Rep to good help!

Reply 1

didgeridoo12uk

the second one is what it wants you to do.

it's basically telling you to change all the x terms to t terms and gives you a relationship how to do it. not really necessary giving you that hint as you spotted it by yourself but some other students won't have.

(edited 13 years ago)

Reply 2

Expendable

Perhaps there was some confusion.

The question was;
"By substituting

t=x^{1/2}

, or otherwise, find the valued of x for which

4x+8=33x^{1/2}

"

So it's implying to change the x^(1/2) to t, but then I would be clueless on the next step. I was saying I would have usually done it by making it into a quadratic equation etc.

As it says or otherwise, does this mean substituting MUST be used to find the correct answer?

Reply 3

AndrewChem

You are on the right track!! If t=x^{1/2} then by squaring both sides you get that t^2 = x
Using these relationships, replace all the x terms, giving you 4t^{2} + 8 = 33t. Make this a quadratic equalling zero. Then either by factorising or using the quadratic formula find solutions. These solutions will be for t, so you need to make them into solutions for x by using the original substitution t=x^{1/2}

Hope that helps!!

Reply 4

bcrazy

Original post by jbeach09

I mean, without substituting, I would have squared the terms and re-arranged to get;

4x^2-33x+8

and then solved some how (factorising/complete square/quad formula) to get x, but it specifically mentions substitution?

+Rep to good help!

AndrewChem is right in his description of how to use the substitution but its is worth noting why this will give a different answer to your method.

You cannot just square each term individually but instead have to "do the same to both sides" i.e. square both sides.

This would leave you with

(4x+8)^2=(33x^{1/2})^2[br]16x^2+64x+64=1089x

which you could then rearrange and solve.
But you can see why the substitution is preferable!

Reply 5

Thrug

It's a hidden quadratic but by telling you t=x^1/2 they are giving you a clue and making it easier.

Reply 6

M-G-C-E

t = x^1/2

4x + 8 = 33x^1/2

therefore:

4x + 8 = 33t

(if t = x^1/2, then t^2 = x)

so,

4t^2 + 8 = 33t

4t^2 - 33t + 8 = 0

(4t - 1)(t - 8)

therefore t = 1/4 or 8

so x = 1/2 or square root 8

Reply 7

shaggybv2

When you have the expression t=1/4 or 8 you would then convert t to x1/2 meaning you would square both sides to find x (x1/2=√x) so x would = 1/16 or 64

Reply 8

TeeEm

Original post by shaggybv2

When you have the expression t=1/4 or 8 you would then convert t to x1/2 meaning you would square both sides to find x (x1/2=√x) so x would = 1/16 or 64