Question 827688
<pre>
Write this

(x  )(x  )

Think of two POSITIVE whole numbers which multiply to give this
number, 24,........

{{{drawing(100,50,-1,10,-30,10,
locate(6.6,2.3,v),locate(6.7,2.6,"|"),
 locate(0,0,x^2-2x-24))}}}

........that subtract (because of this sign)....,

{{{drawing(100,50,-1,10,-30,10,
locate(5.1,2.3,v),locate(5.2,2.6,"|"),
 locate(0,0,x^2-2x-24))}}}

.......to give this POSITIVE number, 2,.......


{{{drawing(100,50,-1,10,-30,10,
locate(3.4,2.3,v),locate(3.5,2.6,"|"),
 locate(0,0,x^2-2x-24))}}}

So the only two POSITIVE integers that multiply
to give 24 and subtract to give 2, are 6 and 4,
because 6 times 4 is 24 and 6 minus 4 is 2.
So we put 6 and 4 in the parentheses like this:

(x 6)(x 4)

All we need do is put the correct signs in the 
middle of each parentheses.

The larger of these two numbers (which is 6 in this case)....
will always get this sign (which is -):


{{{drawing(100,50,-1,10,-30,10,
locate(1.9,2.3,v),locate(2,2.6,"|"),
 locate(0,0,x^2-2x-24))}}}

So we put a "-" sign before the 6 in the
first parentheses:

(x-6)(x 4)

Now we need to decide what sign goes in the
middle of the second parentheses:

We look back at this sign:

{{{drawing(100,50,-1,10,-30,10,
locate(5.1,2.3,v),locate(5.2,2.6,"|"),
 locate(0,0,x^2-2x-24))}}}

Since it is a "-" the second parentheses gets
the OPPOSITE sign from the sign in the other
parentheses (so it gets a + ):

(x-6)(x+4)

That's the answer.

---------------------------------

Let's do another problem like that:

x²-9x+14

Write this

(x  )(x  )

Think of two POSITIVE whole numbers which multiply to give this
number, 14,........

{{{drawing(100,50,-1,10,-30,10,
locate(6.6,2.3,v),locate(6.7,2.6,"|"),
 locate(0,0,x^2-9x+14))}}}

........that add (because of this + sign)....,

{{{drawing(100,50,-1,10,-30,10,
locate(5.1,2.3,v),locate(5.2,2.6,"|"),
 locate(0,0,x^2-9x+14))}}}

.......to give this POSITIVE number, 9,.......


{{{drawing(100,50,-1,10,-30,10,
locate(3.4,2.3,v),locate(3.5,2.6,"|"),
 locate(0,0,x^2-9x-14))}}}

So the only two POSITIVE integers that multiply
to give 14 and add to give 9, are 7 and 2,
because 7 times 2 is 14 and 7 plus 2 is 9.
So we put 7 and 2 in the parentheses like this:

(x 7)(x 2)

All we need do is put the correct signs in the 
middle of each parentheses.

The larger of these two numbers (which is 7 in this case)....
will always get this sign (which is -):


{{{drawing(100,50,-1,10,-30,10,
locate(1.9,2.3,v),locate(2,2.6,"|"),
 locate(0,0,x^2-9x+14))}}}

So we put a "-" sign before the 7 in the
first parentheses:

(x-7)(x 2)

Now we need to decide what sign goes in the
middle of the second parentheses:

We look back at this sign:

{{{drawing(100,50,-1,10,-30,10,
locate(5.1,2.3,v),locate(5.2,2.6,"|"),
 locate(0,0,x^2-9x+14))}}}

Since it is a "+" the second parentheses gets
the SAME sign as the sign in the other
parentheses (so it also gets a - ):

(x-7)(x-2)

That's the answer.

Edwin</pre>