If the distance of point (5, 0) to (3, 2) is twice the distance of (5, 0) to (7, n + 1). Find the value of n.
To find the value of n in this problem, we can start by calculating the distances between the given points.
Let's calculate the distance between (5, 0) and (3, 2) using the distance formula, which is:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the values, we have:
distance1 = sqrt((3 - 5)^2 + (2 - 0)^2)
= sqrt((-2)^2 + 2^2)
= sqrt(4 + 4)
= sqrt(8)
Now, let's calculate the distance between (5, 0) and (7, n + 1):
distance2 = sqrt((7 - 5)^2 + (n + 1 - 0)^2)
= sqrt(2^2 + (n + 1)^2)
= sqrt(4 + n^2 + 2n + 1)
= sqrt(n^2 + 2n + 5)
According to the problem, distance1 is twice distance2:
sqrt(8) = 2 * sqrt(n^2 + 2n + 5)
To solve this equation for n, let's square both sides of the equation:
8 = 4(n^2 + 2n + 5)
Now, expand and simplify the equation:
8 = 4n^2 + 8n + 20
Rearrange the equation to set it to 0:
4n^2 + 8n + 20 - 8 = 0
4n^2 + 8n + 12 = 0
Divide the equation by 4:
n^2 + 2n + 3 = 0
Now, we have a quadratic equation. To solve it, we can use the quadratic formula:
n = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 2, and c = 3. Substituting these values into the quadratic formula, we get:
n = (-2 ± sqrt(2^2 - 4(1)(3))) / (2(1))
n = (-2 ± sqrt(4 - 12)) / 2
n = (-2 ± sqrt(-8)) / 2
n = (-2 ± 2i√2) / 2
n = -1 ± i√2
Therefore, the value of n is either -1 + i√2 or -1 - i√2.