Get 0 on the right by subtracting 192 from both sides

5x^2-68x-192=0

Multiply 5 by -192, getting -960.  Now we make a column of
ways to factor 960 using two integers:

factors
 1x960
 2x480
 3x320
 4x240
 5x192
 6x160
 8x120
10x96
12x80
15x64
16x60
20x48
24x40
30x32

Now since the sign of -960 is negative, we make
a list of the differences, larger minus smaller:

factors   differences, larger - smaller
 1x960           960-1  = 959
 2x480           480-2  = 478
 3x320           320-3  = 317
 4x240           240-4  = 236
 5x192           192-5  = 187 
 6x160           160-6  = 154
 8x120           120-8  = 111
10x96             96-10 = 86
12x80             80-12 = 68
15x64             64-15 = 49 
16x60             60-16 = 44
20x48             48-20 = 28
24x40             40-24 = 16
30x32             32-30 = 2


Now you find the one that agrees in absolute value

with the middle term of 



5x^2-68x-192=0



It is the one I have colored red:

factors   differences, smaller - larger
 1x960           960-1  = 959
 2x480           480-2  = 478
 3x320           320-3  = 317
 4x240           240-4  = 236
 5x192           192-5  = 187 
 6x160           160-6  = 154
 8x120           120-8  = 111
10x96             96-10 = 86
12x80             80-12 = 68
15x64             64-15 = 49 
16x60             60-16 = 44
20x48             48-20 = 28
24x40             40-24 = 16
30x32             32-30 = 2

So we must rewrite the middle coefficient of
5x^2-68x-192=0

which is -68x, using the numbers 12 and 80.

We do that by writing -68x as 12x-80x

So

5x^2-68x-192=0

becomes

5x^2+12x-80x-196=0

Now factor x out of the first two terms:

x(5x+12)-80x-196=0

Now factor -16 out of the last two terms:

x(5x+12)-16(5x+12) = 0

Factor out (5x+12)

(5x+12)(x-16)=0

Use the xero factor property:

5x+12=0          x-16=0
   5x=-12           x=16   
    x=-12/5  

So the solutions are 12/5 and 16

Edwin


 

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