```Question 132417

{{{25w^2- 40w + 16 }}} Rearrange the terms

Looking at {{{25w^2-40w+16}}} we can see that the first term is {{{25w^2}}} and the last term is {{{16}}} where the coefficients are 25 and 16 respectively.

Now multiply the first coefficient 25 and the last coefficient 16 to get 400. Now what two numbers multiply to 400 and add to the  middle coefficient -40? Let's list all of the factors of 400:

Factors of 400:

1,2,4,5,8,10,16,20,25,40,50,80,100,200

-1,-2,-4,-5,-8,-10,-16,-20,-25,-40,-50,-80,-100,-200 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 400

1*400

2*200

4*100

5*80

8*50

10*40

16*25

20*20

(-1)*(-400)

(-2)*(-200)

(-4)*(-100)

(-5)*(-80)

(-8)*(-50)

(-10)*(-40)

(-16)*(-25)

(-20)*(-20)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -40? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -40

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">400</td><td>1+400=401</td></tr><tr><td align="center">2</td><td align="center">200</td><td>2+200=202</td></tr><tr><td align="center">4</td><td align="center">100</td><td>4+100=104</td></tr><tr><td align="center">5</td><td align="center">80</td><td>5+80=85</td></tr><tr><td align="center">8</td><td align="center">50</td><td>8+50=58</td></tr><tr><td align="center">10</td><td align="center">40</td><td>10+40=50</td></tr><tr><td align="center">16</td><td align="center">25</td><td>16+25=41</td></tr><tr><td align="center">20</td><td align="center">20</td><td>20+20=40</td></tr><tr><td align="center">-1</td><td align="center">-400</td><td>-1+(-400)=-401</td></tr><tr><td align="center">-2</td><td align="center">-200</td><td>-2+(-200)=-202</td></tr><tr><td align="center">-4</td><td align="center">-100</td><td>-4+(-100)=-104</td></tr><tr><td align="center">-5</td><td align="center">-80</td><td>-5+(-80)=-85</td></tr><tr><td align="center">-8</td><td align="center">-50</td><td>-8+(-50)=-58</td></tr><tr><td align="center">-10</td><td align="center">-40</td><td>-10+(-40)=-50</td></tr><tr><td align="center">-16</td><td align="center">-25</td><td>-16+(-25)=-41</td></tr><tr><td align="center">-20</td><td align="center">-20</td><td>-20+(-20)=-40</td></tr></table>

From this list we can see that -20 and -20 add up to -40 and multiply to 400

Now looking at the expression {{{25w^2-40w+16}}}, replace {{{-40w}}} with {{{-20w+-20w}}} (notice {{{-20w+-20w}}} adds up to {{{-40w}}}. So it is equivalent to {{{-40w}}})

{{{25w^2+highlight(-20w+-20w)+16}}}

Now let's factor {{{25w^2-20w-20w+16}}} by grouping:

{{{(25w^2-20w)+(-20w+16)}}} Group like terms

{{{5w(5w-4)-4(5w-4)}}} Factor out the GCF of {{{5w}}} out of the first group. Factor out the GCF of {{{-4}}} out of the second group

{{{(5w-4)(5w-4)}}} Since we have a common term of {{{5w-4}}}, we can combine like terms

So {{{25w^2-20w-20w+16}}} factors to {{{(5w-4)(5w-4)}}}

So this also means that {{{25w^2-40w+16}}} factors to {{{(5w-4)(5w-4)}}} (since {{{25w^2-40w+16}}} is equivalent to {{{25w^2-20w-20w+16}}})

note:  {{{(5w-4)(5w-4)}}} is equivalent to  {{{(5w-4)^2}}} since the term {{{5w-4}}} occurs twice. So {{{25w^2-40w+16}}} also factors to {{{(5w-4)^2}}}

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