Question 144730: i need to factor the quadratic expression completely and find the roots of this expression. i came up with (15x-13) & (9x-7) is this correct for this problem? Thank you.
135x2 - 222x + 91
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Looking at  we can see that the first term is  and the last term is  where the coefficients are 135 and 91 respectively.
Now multiply the first coefficient 135 and the last coefficient 91 to get 12285. Now what two numbers multiply to 12285 and add to the middle coefficient -222? Let's list all of the factors of 12285:
Factors of 12285:
1,3,5,7,9,13,15,21,27,35,39,45,63,65,91,105,117,135,189,195,273,315,351,455,585,819,945,1365,1755,2457,4095,12285
-1,-3,-5,-7,-9,-13,-15,-21,-27,-35,-39,-45,-63,-65,-91,-105,-117,-135,-189,-195,-273,-315,-351,-455,-585,-819,-945,-1365,-1755,-2457,-4095,-12285 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 12285
1*12285
3*4095
5*2457
7*1755
9*1365
13*945
15*819
21*585
27*455
35*351
39*315
45*273
63*195
65*189
91*135
105*117
(-1)*(-12285)
(-3)*(-4095)
(-5)*(-2457)
(-7)*(-1755)
(-9)*(-1365)
(-13)*(-945)
(-15)*(-819)
(-21)*(-585)
(-27)*(-455)
(-35)*(-351)
(-39)*(-315)
(-45)*(-273)
(-63)*(-195)
(-65)*(-189)
(-91)*(-135)
(-105)*(-117)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -222? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -222
| First Number | Second Number | Sum | | 1 | 12285 | 1+12285=12286 | | 3 | 4095 | 3+4095=4098 | | 5 | 2457 | 5+2457=2462 | | 7 | 1755 | 7+1755=1762 | | 9 | 1365 | 9+1365=1374 | | 13 | 945 | 13+945=958 | | 15 | 819 | 15+819=834 | | 21 | 585 | 21+585=606 | | 27 | 455 | 27+455=482 | | 35 | 351 | 35+351=386 | | 39 | 315 | 39+315=354 | | 45 | 273 | 45+273=318 | | 63 | 195 | 63+195=258 | | 65 | 189 | 65+189=254 | | 91 | 135 | 91+135=226 | | 105 | 117 | 105+117=222 | | -1 | -12285 | -1+(-12285)=-12286 | | -3 | -4095 | -3+(-4095)=-4098 | | -5 | -2457 | -5+(-2457)=-2462 | | -7 | -1755 | -7+(-1755)=-1762 | | -9 | -1365 | -9+(-1365)=-1374 | | -13 | -945 | -13+(-945)=-958 | | -15 | -819 | -15+(-819)=-834 | | -21 | -585 | -21+(-585)=-606 | | -27 | -455 | -27+(-455)=-482 | | -35 | -351 | -35+(-351)=-386 | | -39 | -315 | -39+(-315)=-354 | | -45 | -273 | -45+(-273)=-318 | | -63 | -195 | -63+(-195)=-258 | | -65 | -189 | -65+(-189)=-254 | | -91 | -135 | -91+(-135)=-226 | | -105 | -117 | -105+(-117)=-222 |
From this list we can see that -105 and -117 add up to -222 and multiply to 12285
Now looking at the expression , replace  with  (notice adds up to . So it is equivalent to )
Now let's factor  by grouping:
 Group like terms
 Factor out the GCF of  out of the first group. Factor out the GCF of  out of the second group
 Since we have a common term of  , we can combine like terms
So factors to
 Set the factorization equal to zero
Now set each factor equal to zero:
 or 
 or  Now solve for x in each case
So our answers are
or
|
|
|
