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The distance between the points (x,21) and (4,7) is 10sqrt(3) Find the sum of all possible values of x.

 Jun 28, 2021
 #1
avatar+234 
+2

Using distance formula,

\(\sqrt{(x-4)^2 + 196} = 10\sqrt{3}\)

Square both sides and subtract 196 on both sides,

\((x-4)^2 + 196 = 300\)

\((x-4)^2 = 104\)

Expand left side and subtract 104 on both sides,

\(x^2 - 8x +16 = 104\)

\(x^2 - 8x -88 = 0\)

Apply quadratic formula to get x values of,

\(x=4±2\sqrt{26}\)

Add both possible values of x together,

\((4+2\sqrt{26}) + (4-2\sqrt{26}) = 8\)

So the final answer is 8.

 Jun 28, 2021
 #3
avatar+2244 
+1

Nice, another approach would be to use Vieta's after finding the quadratic. 

x^2 - 8x - 88, sum of possible values is -b/a. 

-(-8)/1 = 8

 

=^._.^=

catmg  Jun 28, 2021
 #4
avatar+234 
+1

Oh haha that's smart I didn't need to use the quadratic formula lol

Awesomeguy  Jun 28, 2021
 #2
avatar+2244 
+1

(10sqrt(3))^2 = (21-7)^2 + (4-x)^2

You can get this equation by creating a right triangle on a coordinate plane and using Pythagorean Theorum. 

300 = 14^2 + (4-x)^2

 

Can you take it from here?

 

=^._.^=

 Jun 28, 2021
 #5
avatar+121062 
0

THX , Awsomeguy  and catmg   !!!!!

 

Nice answers    !!!!!

 

 

cool cool cool

CPhill  Jun 28, 2021

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