SOLUTION: What is the solution for this? Please use the 4 methods in quadratic equation to see the answer: *extracting the square root *Factoring *Completing the square and *Quadratic F

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: What is the solution for this? Please use the 4 methods in quadratic equation to see the answer: *extracting the square root *Factoring *Completing the square and *Quadratic F      Log On


   

Question 886874: What is the solution for this?
Please use the 4 methods in quadratic equation to see the answer:
*extracting the square root
*Factoring
*Completing the square
and *Quadratic Formula
The length of a rectangular garden is one more than thrice it's width. Its area is 80 m^2. Find the length and the width.
Much thanks!
Found 2 solutions by josgarithmetic, richwmiller:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
Assigning variables,
Length is L, width is w.
L=1%2B3w and wL=80. Substitution for L gives w%283w%2B1%29=80, 3w%5E2%2Bw=80
highlight_green%283w%5E2%2Bw-80=0%29
You can try to factor that. Product -80 will have various combinations but you need to also have
the coefficient of 3 on the leading term.
For 80: 20 & 4, 2 & 40, 8 & 10, ... others.
Easiest to avoid trying to factorize.
-
COMPLETE THE SQUARE-----
First factor the 3.
3%28w%5E2%2B%281%2F3%29w-80%2F3%29=0
The term to use is %281%2F6%29%5E2 (Reading-up on this will help);
3%28w%5E2%2B%281%2F3%29w%2B1%2F36-1%2F36-80%2F3%29=0
3%28%28w%2B1%2F6%29%5E2-%281%2F36%2B80%2F3%29%29=0
3%28%28w%2B1%2F6%29%5E2-%28961%2F36%29%29=0
3%28w%2B1%2F6%29%5E2-961%2F12=0
3%28w%2B1%2F6%29%5E2=961%2F12
%28w%2B1%2F6%29%5E2=961%2F36
w%2B1%2F6=0%2B-+sqrt%28961%2F36%29
w=-1%2F6%2B-+31%2F6
w=%28-1-31%29%2F6 OR w=%28-1%2B31%29%2F6
cross%28w=-32%2F6%29 OR highlight%28w=5%29
The meaningful result here is w=5 for width.

You could have found 5 and 16 for your attempt at factorization of the general quadratic in w.
That seemed like something which would take too many different trys to find.

Now, you can just use the formula for L and the now known value for w.
L=3w%2B1
L=3%2A5%2B1
highlight%28L=16%29

Also, once getting the earlier simpler quadratic in w, the general solution could have been chosen:
3w%5E2%2Bw-80=0;
w=%28-1%2B-+sqrt%281-4%2A3%2A%28-80%29%29%29%2F%282%2A3%29
and then evaluate (and choose the value which makes sense.)

Answer by richwmiller(17219) 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 3w%5E2%2Bw-80, we can see that the first coefficient is 3, the second coefficient is 1, and the last term is -80.



Now multiply the first coefficient 3 by the last term -80 to get %283%29%28-80%29=-240.



Now the question is: what two whole numbers multiply to -240 (the previous product) and add to the second coefficient 1?



To find these two numbers, we need to list all of the factors of -240 (the previous product).



Factors of -240:

1,2,3,4,5,6,8,10,12,15,16,20,24,30,40,48,60,80,120,240

-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-16,-20,-24,-30,-40,-48,-60,-80,-120,-240



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



These factors pair up and multiply to -240.

1*(-240) = -240
2*(-120) = -240
3*(-80) = -240
4*(-60) = -240
5*(-48) = -240
6*(-40) = -240
8*(-30) = -240
10*(-24) = -240
12*(-20) = -240
15*(-16) = -240
(-1)*(240) = -240
(-2)*(120) = -240
(-3)*(80) = -240
(-4)*(60) = -240
(-5)*(48) = -240
(-6)*(40) = -240
(-8)*(30) = -240
(-10)*(24) = -240
(-12)*(20) = -240
(-15)*(16) = -240


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:



First NumberSecond NumberSum
1-2401+(-240)=-239
2-1202+(-120)=-118
3-803+(-80)=-77
4-604+(-60)=-56
5-485+(-48)=-43
6-406+(-40)=-34
8-308+(-30)=-22
10-2410+(-24)=-14
12-2012+(-20)=-8
15-1615+(-16)=-1
-1240-1+240=239
-2120-2+120=118
-380-3+80=77
-460-4+60=56
-548-5+48=43
-640-6+40=34
-830-8+30=22
-1024-10+24=14
-1220-12+20=8
-1516-15+16=1




From the table, we can see that the two numbers -15 and 16 add to 1 (the middle coefficient).



So the two numbers -15 and 16 both multiply to -240 and add to 1



Now replace the middle term 1w with -15w%2B16w. Remember, -15 and 16 add to 1. So this shows us that -15w%2B16w=1w.



3w%5E2%2Bhighlight%28-15w%2B16w%29-80 Replace the second term 1w with -15w%2B16w.



%283w%5E2-15w%29%2B%2816w-80%29 Group the terms into two pairs.



3w%28w-5%29%2B%2816w-80%29 Factor out the GCF 3w from the first group.



3w%28w-5%29%2B16%28w-5%29 Factor out 16 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.



%283w%2B16%29%28w-5%29 Combine like terms. Or factor out the common term w-5



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



Answer:



So 3%2Aw%5E2%2Bw-80 factors to %283w%2B16%29%28w-5%29.



In other words, 3%2Aw%5E2%2Bw-80=%283w%2B16%29%28w-5%29.



Note: you can check the answer by expanding %283w%2B16%29%28w-5%29 to get 3%2Aw%5E2%2Bw-80 or by graphing the original expression and the answer (the two graphs should be identical).