SOLUTION: Please solve by grouping: 63q^2-52q-20

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Question 454099: Please solve by grouping:
63q^2-52q-20
Found 2 solutions by rwm, richard1234:
Answer by rwm(914) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 63q%5E2-52q-20, we can see that the first coefficient is 63, the second coefficient is -52, and the last term is -20.



Now multiply the first coefficient 63 by the last term -20 to get %2863%29%28-20%29=-1260.



Now the question is: what two whole numbers multiply to -1260 (the previous product) and add to the second coefficient -52?



To find these two numbers, we need to list all of the factors of -1260 (the previous product).



Factors of -1260:

1,2,3,4,5,6,7,9,10,12,14,15,18,20,21,28,30,35,36,42,45,60,63,70,84,90,105,126,140,180,210,252,315,420,630,1260

-1,-2,-3,-4,-5,-6,-7,-9,-10,-12,-14,-15,-18,-20,-21,-28,-30,-35,-36,-42,-45,-60,-63,-70,-84,-90,-105,-126,-140,-180,-210,-252,-315,-420,-630,-1260



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



These factors pair up and multiply to -1260.

1*(-1260) = -1260
2*(-630) = -1260
3*(-420) = -1260
4*(-315) = -1260
5*(-252) = -1260
6*(-210) = -1260
7*(-180) = -1260
9*(-140) = -1260
10*(-126) = -1260
12*(-105) = -1260
14*(-90) = -1260
15*(-84) = -1260
18*(-70) = -1260
20*(-63) = -1260
21*(-60) = -1260
28*(-45) = -1260
30*(-42) = -1260
35*(-36) = -1260
(-1)*(1260) = -1260
(-2)*(630) = -1260
(-3)*(420) = -1260
(-4)*(315) = -1260
(-5)*(252) = -1260
(-6)*(210) = -1260
(-7)*(180) = -1260
(-9)*(140) = -1260
(-10)*(126) = -1260
(-12)*(105) = -1260
(-14)*(90) = -1260
(-15)*(84) = -1260
(-18)*(70) = -1260
(-20)*(63) = -1260
(-21)*(60) = -1260
(-28)*(45) = -1260
(-30)*(42) = -1260
(-35)*(36) = -1260


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -52:



First NumberSecond NumberSum
1-12601+(-1260)=-1259
2-6302+(-630)=-628
3-4203+(-420)=-417
4-3154+(-315)=-311
5-2525+(-252)=-247
6-2106+(-210)=-204
7-1807+(-180)=-173
9-1409+(-140)=-131
10-12610+(-126)=-116
12-10512+(-105)=-93
14-9014+(-90)=-76
15-8415+(-84)=-69
18-7018+(-70)=-52
20-6320+(-63)=-43
21-6021+(-60)=-39
28-4528+(-45)=-17
30-4230+(-42)=-12
35-3635+(-36)=-1
-11260-1+1260=1259
-2630-2+630=628
-3420-3+420=417
-4315-4+315=311
-5252-5+252=247
-6210-6+210=204
-7180-7+180=173
-9140-9+140=131
-10126-10+126=116
-12105-12+105=93
-1490-14+90=76
-1584-15+84=69
-1870-18+70=52
-2063-20+63=43
-2160-21+60=39
-2845-28+45=17
-3042-30+42=12
-3536-35+36=1




From the table, we can see that the two numbers 18 and -70 add to -52 (the middle coefficient).



So the two numbers 18 and -70 both multiply to -1260 and add to -52



Now replace the middle term -52q with 18q-70q. Remember, 18 and -70 add to -52. So this shows us that 18q-70q=-52q.



63q%5E2%2Bhighlight%2818q-70q%29-20 Replace the second term -52q with 18q-70q.



%2863q%5E2%2B18q%29%2B%28-70q-20%29 Group the terms into two pairs.



9q%287q%2B2%29%2B%28-70q-20%29 Factor out the GCF 9q from the first group.



9q%287q%2B2%29-10%287q%2B2%29 Factor out 10 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.



%289q-10%29%287q%2B2%29 Combine like terms. Or factor out the common term 7q%2B2



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Answer:



So 63%2Aq%5E2-52%2Aq-20 factors to %289q-10%29%287q%2B2%29.



In other words, 63%2Aq%5E2-52%2Aq-20=%289q-10%29%287q%2B2%29.



Note: you can check the answer by expanding %289q-10%29%287q%2B2%29 to get 63%2Aq%5E2-52%2Aq-20 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
The fastest way is simply to use the quadratic formula. The roots of the equation are



This simplifies to q = -2/7 or q = 10/9. Using the fact that if q is a root of a polynomial, then x-q is a factor of the polynomial, the original polynomial can be expressed as



Usually when we factor something, we want all coefficients to be integers. We can distribute the 63 over the other two factors: