To solve for the value(s) of [tex]\( x \)[/tex] such that the distance between the points [tex]\((18, 3)\)[/tex] and [tex]\((x, -5)\)[/tex] is 17, we use the distance formula:
[tex]\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \text{distance} \][/tex]
Given:
- [tex]\( (x_1, y_1) = (18, 3) \)[/tex]
- [tex]\( (x_2, y_2) = (x, -5) \)[/tex]
- distance = 17
First, plug in the given coordinates and the distance into the distance formula:
[tex]\[ \sqrt{(x - 18)^2 + (-5 - 3)^2} = 17 \][/tex]
Simplify inside the square root:
[tex]\[ \sqrt{(x - 18)^2 + (-8)^2} = 17 \][/tex]
[tex]\[ \sqrt{(x - 18)^2 + 64} = 17 \][/tex]
Next, square both sides to eliminate the square root:
[tex]\[ (x - 18)^2 + 64 = 17^2 \][/tex]
[tex]\[ (x - 18)^2 + 64 = 289 \][/tex]
Now, isolate the [tex]\((x - 18)^2\)[/tex] term:
[tex]\[ (x - 18)^2 = 289 - 64 \][/tex]
[tex]\[ (x - 18)^2 = 225 \][/tex]
To solve for [tex]\( x \)[/tex], take the square root of both sides:
[tex]\[ x - 18 = \pm 15 \][/tex]
This gives us two possible equations:
1. [tex]\( x - 18 = 15 \)[/tex]
2. [tex]\( x - 18 = -15 \)[/tex]
Solve each equation for [tex]\( x \)[/tex]:
1. [tex]\( x - 18 = 15 \)[/tex]
[tex]\[ x = 33 \][/tex]
2. [tex]\( x - 18 = -15 \)[/tex]
[tex]\[ x = 3 \][/tex]
Thus, the values of [tex]\( x \)[/tex] that satisfy the given condition are:
[tex]\[ x = 33, 3 \][/tex]
So, the answer is:
[tex]\[ x = 33, 3 \][/tex]
