document.write( "Question 817230: I really have trouble with word problems! I'd love some help. Thanks! :)
\n" ); document.write( "One number is 3 less than twice another. If their product is 405, find the numbers.
\n" ); document.write( "I tried: x=3-2x, 3-2x=x ??
\n" ); document.write( "I'm just not sure how to do this. \n" ); document.write( "
Algebra.Com's Answer #491956 by richwmiller(17219)\"\" \"About 
You can put this solution on YOUR website!
you have two numbers
\n" ); document.write( "one number is x \r
\n" ); document.write( "\n" ); document.write( "is 3 less than -3
\n" ); document.write( "twice another \r
\n" ); document.write( "\n" ); document.write( "and another is y
\n" ); document.write( "2y twice another
\n" ); document.write( "x=2y-3\r
\n" ); document.write( "\n" ); document.write( "their product means multiply them
\n" ); document.write( "xy=405
\n" ); document.write( "now we have
\n" ); document.write( "x=2y-3
\n" ); document.write( "xy=405
\n" ); document.write( "x=405/y
\n" ); document.write( "both equations =x so they equal each other
\n" ); document.write( "2y-3=405/y
\n" ); document.write( "2y^2-3y=405
\n" ); document.write( "2y^2-3y-405=0
\n" ); document.write( "find factors of -810 (2*-405) that add up to -3
\n" ); document.write( "2y^2-
\n" ); document.write( "(y-15) (2 y+27) = 0
\n" ); document.write( "y = 15 y = -27/2
\n" ); document.write( "x=27
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression \"2y%5E2-3y-405\", we can see that the first coefficient is \"2\", the second coefficient is \"-3\", and the last term is \"-405\".



Now multiply the first coefficient \"2\" by the last term \"-405\" to get \"%282%29%28-405%29=-810\".



Now the question is: what two whole numbers multiply to \"-810\" (the previous product) and add to the second coefficient \"-3\"?



To find these two numbers, we need to list all of the factors of \"-810\" (the previous product).



Factors of \"-810\":

1,2,3,5,6,9,10,15,18,27,30,45,54,81,90,135,162,270,405,810

-1,-2,-3,-5,-6,-9,-10,-15,-18,-27,-30,-45,-54,-81,-90,-135,-162,-270,-405,-810



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to \"-810\".

1*(-810) = -810
2*(-405) = -810
3*(-270) = -810
5*(-162) = -810
6*(-135) = -810
9*(-90) = -810
10*(-81) = -810
15*(-54) = -810
18*(-45) = -810
27*(-30) = -810
(-1)*(810) = -810
(-2)*(405) = -810
(-3)*(270) = -810
(-5)*(162) = -810
(-6)*(135) = -810
(-9)*(90) = -810
(-10)*(81) = -810
(-15)*(54) = -810
(-18)*(45) = -810
(-27)*(30) = -810


Now let's add up each pair of factors to see if one pair adds to the middle coefficient \"-3\":



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First NumberSecond NumberSum
1-8101+(-810)=-809
2-4052+(-405)=-403
3-2703+(-270)=-267
5-1625+(-162)=-157
6-1356+(-135)=-129
9-909+(-90)=-81
10-8110+(-81)=-71
15-5415+(-54)=-39
18-4518+(-45)=-27
27-3027+(-30)=-3
-1810-1+810=809
-2405-2+405=403
-3270-3+270=267
-5162-5+162=157
-6135-6+135=129
-990-9+90=81
-1081-10+81=71
-1554-15+54=39
-1845-18+45=27
-2730-27+30=3




From the table, we can see that the two numbers \"27\" and \"-30\" add to \"-3\" (the middle coefficient).



So the two numbers \"27\" and \"-30\" both multiply to \"-810\" and add to \"-3\"



Now replace the middle term \"-3y\" with \"27y-30y\". Remember, \"27\" and \"-30\" add to \"-3\". So this shows us that \"27y-30y=-3y\".



\"2y%5E2%2Bhighlight%2827y-30y%29-405\" Replace the second term \"-3y\" with \"27y-30y\".



\"%282y%5E2%2B27y%29%2B%28-30y-405%29\" Group the terms into two pairs.



\"y%282y%2B27%29%2B%28-30y-405%29\" Factor out the GCF \"y\" from the first group.



\"y%282y%2B27%29-15%282y%2B27%29\" Factor out \"15\" from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



\"%28y-15%29%282y%2B27%29\" Combine like terms. Or factor out the common term \"2y%2B27\"



===============================================================



Answer:



So \"2%2Ay%5E2-3%2Ay-405\" factors to \"%28y-15%29%282y%2B27%29\".



In other words, \"2%2Ay%5E2-3%2Ay-405=%28y-15%29%282y%2B27%29\".



Note: you can check the answer by expanding \"%28y-15%29%282y%2B27%29\" to get \"2%2Ay%5E2-3%2Ay-405\" or by graphing the original expression and the answer (the two graphs should be identical).

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