To find the value of [tex]\( x \)[/tex] such that the distance between the points [tex]\((x, 3)\)[/tex] and [tex]\((-x, 2)\)[/tex] is 5, we can use the distance formula. Here's a step-by-step solution:
1. Set up the distance formula:
The distance [tex]\( d \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
2. Plug in the coordinates:
In our case, the points are [tex]\((x, 3)\)[/tex] and [tex]\((-x, 2)\)[/tex], and the distance [tex]\( d \)[/tex] is given as 5. So,
[tex]\[
5 = \sqrt{((-x) - x)^2 + (2 - 3)^2}
\][/tex]
3. Simplify inside the square root:
[tex]\[
5 = \sqrt{(-2x)^2 + (-1)^2}
\][/tex]
[tex]\[
5 = \sqrt{4x^2 + 1}
\][/tex]
4. Square both sides to eliminate the square root:
[tex]\[
5^2 = (4x^2 + 1)
\][/tex]
[tex]\[
25 = 4x^2 + 1
\][/tex]
5. Solve for [tex]\( x^2 \)[/tex]:
Subtract 1 from both sides:
[tex]\[
25 - 1 = 4x^2
\][/tex]
[tex]\[
24 = 4x^2
\][/tex]
Divide both sides by 4:
[tex]\[
x^2 = 6
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Take the square root of both sides:
[tex]\[
x = \pm\sqrt{6}
\][/tex]
So, the values of [tex]\( x \)[/tex] that satisfy the given condition are [tex]\( x = \sqrt{6} \)[/tex] and [tex]\( x = -\sqrt{6} \)[/tex].
Given the multiple-choice options, it appears there might be an error or misrepresentation in the options listed. Correctly, [tex]\( x \)[/tex] should be either [tex]\( \sqrt{6} \approx 2.449 \)[/tex] or [tex]\(-\sqrt{6} \approx -2.449 \)[/tex]. None of the provided choices match this exactly.
Thus, the correct answers are:
[tex]\[
x = \sqrt{6} \approx 2.45 \quad \text{or} \quad x = -\sqrt{6} \approx -2.45
\][/tex]
