Factorising

With those, I think you just have to 'see' it. Maybe someone has a better method (EDIT: Turns out Maths12345 does ), but if you can't see it then I'd just complete the square.

First thing to always check is if you can simplify the equation. In this case you can't.

16x^2+16x-45

The product is a*c: 16*-45=-720
The sum is b: +16

What 2 numbers give a product of -720 and a sum of +16?

36 and -20.

Now split the b term into this: (you should put it in a order, so you can factorise e.g. 16 and 36 (have 4 in common) and 20 and 45 (have 5 in common).

16x^2+36x-20x-45

Factorise


4x(4x+9)-5(4x+9)

Common factor?

(4x+9)

Next bracket is the bits left over.

(4x-5)

Answer:

(4x+9)(4x-5)

Thanks to you both for the help.

The bolded bit is the main problem I have so I guess hassi94 is, sadly , right.

Do you guys have any ideas on how to improve on coming up with the two required numbers? Do I just need to get more a feel for advanced times tables?! Can't say I want to learn everything up to and including 36x36 just in case things like this come up. Would you halve 720 to see that it's 36x10 and then just 'see' that 36 and 20 are right?

Hopefully nothing too ridiculous comes up in the actual exam. This is frustrating.

Thanks to you both for the help.

The bolded bit is the main problem I have so I guess hassi94 is, sadly , right.

Do you guys have any ideas on how to improve on coming up with the two required numbers? Do I just need to get more a feel for advanced times tables?! Can't say I want to learn everything up to and including 36x36 just in case things like this come up. Would you halve 720 to see that it's 36x10 and then just 'see' that 36 and 20 are right?

Hopefully nothing too ridiculous comes up in the actual exam. This is frustrating.

If you had $y+3y^2$ instead, would you be able to see what to do?

The only sure fire way of getting the solution is by doing it algebraically (e.g. a+b=16, ab=-720 and solve simultaneously) but this will just end up with another quadratic that would probably be no easier to solve

I do highly recommend just completing the square if you get stuck (or even the quadratic formula - but completing the square will more often come up handy in future mathematics I think), there's very little in the way of having to 'spot' anything - you just do it

But resorting to completing the square or using the quadratic formula is tantamount to admitting defeat!

Haha well the most I can say then is just get used to it; practice several questions.

27x^2 + 15x - 22 = 0

Factorise that for fun!

Took like 3-4 mins of checking numbers on the calc but I did it!



Any more tricky ones so I can practise and ask you if I get stuck?

Took like 3-4 mins of checking numbers on the calc but I did it!



Any more tricky ones so I can practise and ask you if I get stuck?

Actually it doesn't takes so much time to solve it.

Here's how i approach such questions:
$27x^2+15x-22=0$

$27 \times 22= 594$

$\begin{array}{c|c} 2 & 594 \\ \hline 3 & 297 \\ \hline 3 & 99 \\ \hline & 33 \end{array}$

So we can notice that, $3 \times 3 \times 2 \times 33 = 594 \ \ and \ \ 33-(3 \times 3 \times 2)=33-18=15$

So
$27x^2+15x-22=0 \implies 27x^2+33x-18x-22=0 \\ \implies 3x(9x+11)-2(9x+11)=0 \implies (3x-2)(9x+11)=0$

Looks like a really good method. Will definitely try it that way when practising and see how it goes.

This is the best method in my opinion, i always use it. It is quite faster and does gives an answer if the expression is factorisable. By other ways it is too difficult to ensure that can the expression be factorised or not.

Seems like too much work to me haha I've never used any 'method' except just 'by inspection'.

Try it, it is very quick.

It looks too much work, because for showing the method i had to do long working which i normally wouldn't do.

If a quadratic ever comes up that I feel I need it for I might do so; but that seems even more unlikely considering I've basically finished A-level Maths/Further (only M2 left and that won't have any in ) - and they don't bother throwing big numbers at you for no real reason at uni/STEP

How's your STEP preparation going?

Is STEP very hard?

It's getting better - I'm feeling a lot more confident and if I improve my question-picking I'm pretty confident on getting my grade in STEP II at least. STEP III I've still got a fair bit to go but I'm not doing bad at all.

It is quite difficult but it just takes a lot of perseverance I think. Obviously not just anyone can persevere and be successful but if you have some problem-solving ability (which you do) then it definitely is possible with practice. I've been doing questions several times per week since January and only just now feel that I'm starting to crack it (with help from the STEP easter school)