First determine whether there are rational factors.
a = 12" "b = -37 " "c = -780a=12 b=−37 c=−780
b^2 - 4ac = (-37)^2 -4(12)(-780) = 38809 = 197^2b2−4ac=(−37)2−4(12)(−780)=38809=1972
We need to find factors of 12 and 78012and780 whose products differ by 3737
The smaller the value if bb, the closer the factors are to sqrt(ac)√ac
sqrt(12 xx 780) = sqrt9360 = 96.7 ~~97√12×780=√9360=96.7≈97
The factors will be on either side of 96.796.7, about 37/2372 bigger or smaller.
37/2 = 18.5372=18.5
97 -18 =79 and 97+18 = 11597−18=79and97+18=115
Use trial and error with values close to 79 and 11579and115
93609360 is not divisible by 7979
9360 div 80 = 117" "9360÷80=117 BINGO! " "117-80 = 37 117−80=37
These are the factors, now we need to form them from 12 and 78012and780
" "12 and 780 12and780
" "darrcolor(white)(xxx)darr ↓×x↓
" "3color(white)(xxxx)20" "rarr 4 xx20 = 80 3××20 →4×20=80
" "4color(white)(xxxx)39" "rarr 3xx39 = ul117
color(white)(xxxxxxxxxxxxxxxxxxxxx)37
These are the factors, now fill in the signs to get -37 and -780
" "12 and 780
" "darrcolor(white)(xxx)darr
" "3color(white)(xxxx)+20" "rarr 4 xx+20 = +80
" "4color(white)(xxxx)-39" "rarr 3xx-39 = -ul117
color(white)(xxxx.xxxxxxxxxxxxxxxxxxxx)-37
(3x+20)(4x-39)=0
Set each factor equal to 0
3x+20 =0 " "rarr x =-20/3
4x-39 =0" "rarr x = 39/4
