document.write( "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.
\n" ); document.write( "135x2 - 222x + 91 \n" ); document.write( "

Algebra.Com's Answer #105466 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at $\"135x%5E2-222x%2B91\"$ we can see that the first term is $\"135x%5E2\"$ and the last term is $\"91\"$ where the coefficients are 135 and 91 respectively.\r
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\n" ); document.write( "\n" ); document.write( "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:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 12285:\r
\n" ); document.write( "\n" ); document.write( "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\r
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\n" ); document.write( "\n" ); document.write( "-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\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to 12285\r
\n" ); document.write( "\n" ); document.write( "1*12285\r
\n" ); document.write( "\n" ); document.write( "3*4095\r
\n" ); document.write( "\n" ); document.write( "5*2457\r
\n" ); document.write( "\n" ); document.write( "7*1755\r
\n" ); document.write( "\n" ); document.write( "9*1365\r
\n" ); document.write( "\n" ); document.write( "13*945\r
\n" ); document.write( "\n" ); document.write( "15*819\r
\n" ); document.write( "\n" ); document.write( "21*585\r
\n" ); document.write( "\n" ); document.write( "27*455\r
\n" ); document.write( "\n" ); document.write( "35*351\r
\n" ); document.write( "\n" ); document.write( "39*315\r
\n" ); document.write( "\n" ); document.write( "45*273\r
\n" ); document.write( "\n" ); document.write( "63*195\r
\n" ); document.write( "\n" ); document.write( "65*189\r
\n" ); document.write( "\n" ); document.write( "91*135\r
\n" ); document.write( "\n" ); document.write( "105*117\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-12285)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-4095)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-2457)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(-1755)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(-1365)\r
\n" ); document.write( "\n" ); document.write( "(-13)*(-945)\r
\n" ); document.write( "\n" ); document.write( "(-15)*(-819)\r
\n" ); document.write( "\n" ); document.write( "(-21)*(-585)\r
\n" ); document.write( "\n" ); document.write( "(-27)*(-455)\r
\n" ); document.write( "\n" ); document.write( "(-35)*(-351)\r
\n" ); document.write( "\n" ); document.write( "(-39)*(-315)\r
\n" ); document.write( "\n" ); document.write( "(-45)*(-273)\r
\n" ); document.write( "\n" ); document.write( "(-63)*(-195)\r
\n" ); document.write( "\n" ); document.write( "(-65)*(-189)\r
\n" ); document.write( "\n" ); document.write( "(-91)*(-135)\r
\n" ); document.write( "\n" ); document.write( "(-105)*(-117)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "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\r
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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

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -105 and -117 add up to -222 and multiply to 12285\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"135x%5E2-222x%2B91\"$ , replace $\"-222x\"$ with $\"-105x%2B-117x\"$ (notice $\"-105x%2B-117x\"$ adds up to $\"-222x\"$ . So it is equivalent to $\"-222x\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"135x%5E2%2Bhighlight%28-105x%2B-117x%29%2B91\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"135x%5E2-105x-117x%2B91\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28135x%5E2-105x%29%2B%28-117x%2B91%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"15x%289x-7%29-13%289x-7%29\"$ Factor out the GCF of $\"15x\"$ out of the first group. Factor out the GCF of $\"-13\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%2815x-13%29%289x-7%29\"$ Since we have a common term of $\"9x-7\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"135x%5E2-222x%2B91\"$ factors to $\"%2815x-13%29%289x-7%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%2815x-13%29%289x-7%29=0\"$ Set the factorization equal to zero\r
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\n" ); document.write( "\n" ); document.write( "Now set each factor equal to zero:\r
\n" ); document.write( "\n" ); document.write( " $\"15x-13=0\"$ or $\"9x-7=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x=13%2F15\"$ or $\"x=7%2F9\"$ Now solve for x in each case\r
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\n" ); document.write( "\n" ); document.write( "So our answers are\r
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\n" ); document.write( "\n" ); document.write( " $\"x=13%2F15\"$ or $\"x=7%2F9\"$ \r
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