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# Inequality problem

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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
+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
+2104
+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
+234
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Oh haha that's smart I didn't need to use the quadratic formula lol

Awesomeguy  Jun 28, 2021
#2
+2104
+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
+120966
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THX , Awsomeguy  and catmg   !!!!!