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  (in our case ) has the following solutons:

  

  

  

  For these solutions to exist, the discriminant  should not be a negative number.

  

  First, we need to compute the discriminant : .

  

  Discriminant d=73 is greater than zero. That means that there are two solutions: .

  

    

    

    

    Quadratic expression  can be factored:

  

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



  

Answer by richwmiller(17219)   (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 , we can see that the first coefficient is , the second coefficient is , and the last term is .



Now multiply the first coefficient  by the last term  to get .



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



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



Factors of :

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 .

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 :



First Number
Second Number
Sum
1
-60
1+(-60)=-59
2
-30
2+(-30)=-28
3
-20
3+(-20)=-17
4
-15
4+(-15)=-11
5
-12
5+(-12)=-7
6
-10
6+(-10)=-4
-1
60
-1+60=59
-2
30
-2+30=28
-3
20
-3+20=17
-4
15
-4+15=11
-5
12
-5+12=7
-6
10
-6+10=4




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



So the two numbers  and  both multiply to  and add to 



Now replace the middle term  with . Remember,  and  add to . So this shows us that .



 Replace the second term  with .



 Group the terms into two pairs.



 Factor out the GCF  from the first group.



 Factor out  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.



 Combine like terms. Or factor out the common term 



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



Answer:



So  factors to .



In other words, .



Note: you can check the answer by expanding  to get  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 

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