SOLUTION: Two cars leave an intersection, one traveling west and the other south. After some time, the faster car is 7 miles farther away from the intersection than the slower car. At that t

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Question 900760: Two cars leave an intersection, one traveling west and the other south. After some time, the faster car is 7 miles farther away from the intersection than the slower car. At that time, the two cars are 13 miles apart. How far did each car travel?
My work: x^2+(x+7)^2=13^2
x^2+x^2+(x+7)(x+7)=169
2x^2+7x+7x=169
2x^2+14x=169
2x^2+14x-155=0
Found 2 solutions by ewatrrr, richwmiller:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
 

Hi 

x^2+(x+7)^2=13^2      Note: (x+7)^2 = (x+7)(x+7) =  x^2 + 14x + 49

 x^2+ x^2 + 14x + 49 =169

2x^2 + 14x = 169-49

2x^2 + 14x = 120

x^2 + 7x - 60 = 0

 


 
Solved by pluggable solver: SOLVE quadratic equation with variable


Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B7x%2B-6+=+0) has the following solutons:

  

  x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

  

  For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

  

  First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%287%29%5E2-4%2A1%2A-6=73.

  

  Discriminant d=73 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-7%2B-sqrt%28+73+%29%29%2F2%5Ca.

  

    x%5B1%5D+=+%28-%287%29%2Bsqrt%28+73+%29%29%2F2%5C1+=+0.772001872658765

    x%5B2%5D+=+%28-%287%29-sqrt%28+73+%29%29%2F2%5C1+=+-7.77200187265877

    

    Quadratic expression 1x%5E2%2B7x%2B-6 can be factored:

  1x%5E2%2B7x%2B-6+=+1%28x-0.772001872658765%29%2A%28x--7.77200187265877%29

  Again, the answer is: 0.772001872658765, -7.77200187265877.
Here's your graph:

graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B7%2Ax%2B-6+%29

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
x^2+(x+7)^2=13^2
x^2+(x+7)(x+7)=169
x^2+x^2+14x+49=169
2x^2+14x=120
2x^2+14x-120=0
x^2+7x-60=0
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2%2B7x-60, we can see that the first coefficient is 1, the second coefficient is 7, and the last term is -60.



Now multiply the first coefficient 1 by the last term -60 to get %281%29%28-60%29=-60.



Now the question is: what two whole numbers multiply to -60 (the previous product) and add to the second coefficient 7?



To find these two numbers, we need to list all of the factors of -60 (the previous product).



Factors of -60:

1,2,3,4,5,6,10,12,15,20,30,60

-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60



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



These factors pair up and multiply to -60.

1*(-60) = -60
2*(-30) = -60
3*(-20) = -60
4*(-15) = -60
5*(-12) = -60
6*(-10) = -60
(-1)*(60) = -60
(-2)*(30) = -60
(-3)*(20) = -60
(-4)*(15) = -60
(-5)*(12) = -60
(-6)*(10) = -60


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:



First NumberSecond NumberSum
1-601+(-60)=-59
2-302+(-30)=-28
3-203+(-20)=-17
4-154+(-15)=-11
5-125+(-12)=-7
6-106+(-10)=-4
-160-1+60=59
-230-2+30=28
-320-3+20=17
-415-4+15=11
-512-5+12=7
-610-6+10=4




From the table, we can see that the two numbers -5 and 12 add to 7 (the middle coefficient).



So the two numbers -5 and 12 both multiply to -60 and add to 7



Now replace the middle term 7x with -5x%2B12x. Remember, -5 and 12 add to 7. So this shows us that -5x%2B12x=7x.



x%5E2%2Bhighlight%28-5x%2B12x%29-60 Replace the second term 7x with -5x%2B12x.



%28x%5E2-5x%29%2B%2812x-60%29 Group the terms into two pairs.



x%28x-5%29%2B%2812x-60%29 Factor out the GCF x from the first group.



x%28x-5%29%2B12%28x-5%29 Factor out 12 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.



%28x%2B12%29%28x-5%29 Combine like terms. Or factor out the common term x-5



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



Answer:



So x%5E2%2B7%2Ax-60 factors to %28x%2B12%29%28x-5%29.



In other words, x%5E2%2B7%2Ax-60=%28x%2B12%29%28x-5%29.



Note: you can check the answer by expanding %28x%2B12%29%28x-5%29 to get x%5E2%2B7%2Ax-60 or by graphing the original expression and the answer (the two graphs should be identical).


x+12=0
x=-12 reject negative distance
x-5=0
x=5 miles
x+7=12 miles
The famous 5,12,13 right triangle
BTW The other tutor entered -6 instead of -60 into the solver