To find the value of a, we need to determine the two points of intersection between the two functions. Let's call the two points of intersection (x1,y1) and (x2,y2).
Setting y = 3x + a equal to y = x^2 + x, we get:
3x + a = x^2 + x
3x + a - x^2 - x = 0
x^2 + 2x + (3-a) = 0
We can now use the quadratic formula to find the solutions for x:
x = (-b ± √(b^2 - 4ac)) / 2a
x = (-2 ± √(2^2 - 4 * 1 * (3-a))) / 2 * 1
x = (-2 ± √(4 + 4a - 4 * 3)) / 2
x = (-2 ± √(4a - 8)) / 2
Since the two points of intersection are distinct, the square root must be real and nonnegative, so:
4a - 8 ≥ 0
a ≥ 2
The two points of intersection can now be found by substituting the values of x back into y = 3x + a:
y1 = 3x1 + a = 3 * (-2 + √(4a - 8)) / 2 + a = -3 + 3√(4a - 8) + a
y2 = 3x2 + a = 3 * (-2 - √(4a - 8)) / 2 + a = -3 - 3√(4a - 8) + a
Finally, the distance between the two points is given as 3 * sqrt(15), so:
√((x1 - x2)^2 + (y1 - y2)^2) = 3 * √(30)
√((-2 + 2√(4a - 8))^2 + (3 + 3√(4a - 8) - (-3 - 3√(4a - 8)))^2) = 3 * √(30)
√(4 + 6√(4a - 8)) = 3 * √(30)
4 + 6√(4a - 8) = 15
6√(4a - 8) = 11
√(4a - 8) = 11 / 6
√(4a - 8) = √(11 / 6)
4a - 8 = 11 / 6 * 11 / 6
4a = 11 / 6 * 11 / 6 + 8
4a = 121 / 36 + 144 / 36
4a = 265 / 36
a = 265 / 36 * 9 / 4
a = 765 / 144
So the value of a is 765/144. This is already in the form of a common fraction, so this is our final answer.
