SOLUTION: 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 - 2

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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 NumberSecond NumberSum
1122851+12285=12286
340953+4095=4098
524575+2457=2462
717557+1755=1762
913659+1365=1374
1394513+945=958
1581915+819=834
2158521+585=606
2745527+455=482
3535135+351=386
3931539+315=354
4527345+273=318
6319563+195=258
6518965+189=254
9113591+135=226
105117105+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

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