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agentmulder

Active member
Moderator edit: This commentary topic pertains to the following tutorial:

I agree with everything MarkFL has posted so i'll just give a quick 'scheme' leaving out the details.

Another approach is to factor by grouping, consider,

$6x^2 - 17x - 45$

$(6)(-45) = -270$

Now find 2 integers that multiply to -270 and sum to -17,

-27 with 10 work

rewrite $-17x$ as $-27x + 10x$

$6x^2 - 27x + 10x - 45$

use grouping, paying mind to the signs,

$(6x^2 - 27x) + (10x -45)$

factor,

$3x(2x - 9) + 5(2x - 9)$

$(2x - 9)(3x + 5)$

You can multiply this out using FOIL to confirm the factoring is correct. Last edited by a moderator:

mathmaniac

Well-known member

$6x^2 - 17x - 45$

$(6)(-45) = -270$

Now find 2 integers that multiply to -270 and sum to -17,

-27 with 10 work
How to do it:

We have to find p and q such that pq=6 x (-45)= 2 x 3 x 3 x 3 x 5

and p+q=-17

We observe that 17 is a prime.So p and q cannot have common factors (otherwise p+q will be have a common factor).

So either p or q will have all the 3 "3"s.Let it be p.
We have a 2 and 5 remaining which can go into either p or q.Let us first try both of them in q.
So we have p=27 and q=10

One and only one of p and q is negative and also the greater of p and q has to have the sign "-" (because the sum is negative).

So we have p + q = -27 + 10 = -17 and it works.If it hadn't worked then we had to exchange the 2 and 5 a couple of times until we get it right.

If p + q had a common factor,then we had to allow it to be the GCD of p and q and continue this.

agentmulder

Active member

How to do it:

We have to find p and q such that pq=6 x (-45)= 2 x 3 x 3 x 3 x 5

and p+q=-17

We observe that 17 is a prime.So p and q cannot have common factors (otherwise p+q will be have a common factor).

So either p or q will have all the 3 "3"s.Let it be p.
We have a 2 and 5 remaining which can go into either p or q.Let us first try both of them in q.
So we have p=27 and q=10

One and only one of p and q is negative and also the greater of p and q has to have the sign "-" (because the sum is negative).

So we have p + q = -27 + 10 = -17 and it works.If it hadn't worked then we had to exchange the 2 and 5 a couple of times until we get it right.

If p + q had a common factor,then we had to allow it to be the GCD of p and q and continue this.
These are nice observations, i especially like the part of all 3's must belong to p or q. As the product AC gets large your observations can in fact reduce the work involved trying to find p and q. Petrus

Well-known member
I can add a method Let's say we want to factor $$\displaystyle p(x)=-4x^2+24x-32$$ we can use this formula $$\displaystyle p(x)=k*(x-a)(x-b)$$
we want to calculate crittical point (without derivate). In Sweden we learned a formula caled "pq-formula"(too calculate critical point in second polynom) as you can see there cant be any number or negative number infront of $$\displaystyle x^2$$ so we start to break out -4 then we got $$\displaystyle -4(x^2-6x+8)$$
now we calculate the critical point $$\displaystyle x^2-6x+8=0$$ and we get $$\displaystyle x_1=4$$ and $$\displaystyle x_2=2$$ now we use the formula $$\displaystyle p(x)=k*(x-a)(x-b)$$ where k is our constant and a,b our crit point so we got $$\displaystyle p(x)=-4*(x-4)(x-2)$$ 