Hi guys,
I've been trying to factorise the following equation:
4p^2 + 12p + 9 = 0
I can't seem to find any values for a and b so that 4a+b=12 and a*b=9
Any help on this?
thanks
I find that explaining methods of factoring is very tricky... basically I take advantage of the fact that I had it drilled into me and I am quite quick with multiplying in my head
But here's my shot at it
$\displaystyle 4x^2+12x+9$
4 has the following factors 1,2,2,4
So if this thing is factorable it has one of the two forms below
1) (x )(4x ) or
2) (2x )(2x )
So I just pick one and run with it, let's say we guess wrong
Try (x )(4x )
Now 9 has the factors 1,3,3,9 so we can put 1,9 together in any order or 3,3 together
Since 9 is positive, either both numbers are negative or both numbers are positive
So our possible combinations are: (x+3)(4x+3), (x-3)(4x-3), (x+1)(4x+9), (x+9)(4x+1), (x-1)(4x-9), (x-9)(4x-1)
I run through all these in my head and realize that I was not getting 12x out of the deal, so I move to the next one
Try (2x )(2x )
Now for blank space we have exactly the same combinations as above, I run through them in my head (and by this I mean I say to myself (2x+1)(2x+9) will give me 20x, nope no good try again)
and I get that (2x+3)(2x+3) is my answer
hope this helps
and by the way, this method is significantly quicker than setting up those equations and looking at it once you get good with this
Ok, I kind of understand it. I also used the quadratic formula and got -3/2 which makes sense:
(x+3/2)(x+3/2) /// multiplied by 2
(2x+3)(2x+3)
So in this case there's only one solution: x=-3/2
I'll need to practise it a lot more. I am ok when the coefficient 'a' is 1 as in x^2+4x........ but when 'a' is different to 1 then I sometimes struggle.
thanks
you can of course always apply the quadratic formula
When you apply it, you get 2 answers in general, call them $\displaystyle r_1$ and $\displaystyle r_2$
Then the quadratic can be written as $\displaystyle (x-r_1)(x-r_2)$
In your case $\displaystyle r_1=r_2=-\frac{3}{2}$
So if you were stuck on a test and had time this is a fail safe, but obviously factoring is preferred
this is a complex trinomal, this means you have to multiply the A value with the C value and find two numbers that add up to 12 and multiply to 36. In this case it will be 6p and 6p. From there to find the great common factor between the first two values and the next two values. If you did it right you will get one value twice in your work.
This is how your work should work:
$\displaystyle 0 = 4p^2+12p+9$
$\displaystyle 0 = 4p^2+6p+6p+9$
$\displaystyle 0 = 2p(2p+3)+3(2p+3)$
$\displaystyle 0 = (2p+3)(2p+3)$
$\displaystyle 0 = (2p+3)^2$
Therefore $\displaystyle (2p+3)^2$ is $\displaystyle 4p^2+12p+9=0$.
P cannot equal -3 over 2. The root is -3 over 2.