Since the parabola has a vertical line of symmetry at x=2, the x-coordinate of the vertex of the parabola is 2. Since the parabola goes through the points (1,1) and (4,-7), we know that the y-coordinate of the vertex is the average of the y-coordinates of these two points:
(1+(-7))/2 = -3
Therefore, the vertex of the parabola is at (2,-3).
Since the vertex is on the parabola, we can plug in the values of the vertex into the equation of the parabola:
a(2)^2 + b(2) + c = -3
4a + 2b + c = -3
Using the points (1,1) and (4,-7) to form two equations:
a(1)^2 + b(1) + c = 1
a(4)^2 + b(4) + c = -7
This simplifies to:
a + b + c = 1
16a + 4b + c = -7
Subtracting the first equation from the second equation:
15a + 3b = -8
5a + b = -8/3
b = -8/3 - 5a
Substitute the value of b back into the first equation:
a - 8/3 - 5a + c = 1
-4a + c = 11/3
Substitute the value of c into the equation 4a + 2b + c = -3:
4a + 2(-8/3 - 5a) + 11/3 = -3
4a - 16/3 - 10a + 11/3 = -3
-6a - 5 = -3
-6a = 2
a = -1/3
Now that we have the value of a, we can substitute it back into the equation for b:
b = -8/3 - 5(-1/3)
b = -8/3 + 5/3
b = -3/3
b = -1
Finally, substitute the values of a and b into one of the original equations to solve for c:
a + b + c = 1
-1/3 - 1 + c = 1
c = 1 + 1 + 1/3
c = 7/3
Now, we have the values of a, b, and c. Since n is the greater root of the quadratic, we can use the quadratic formula to find the roots:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the values of a, b, and c, we get the roots:
x = (-(-1) ± √((-1)^2 - 4(-1/3)(7/3))) / (2(-1))
x = (1 ± √(1 + 28/3)) / -2
x = (1 ± √(31/3)) / -2
Since n is the greater root, n = (1 + √(31/3)) / -2
Therefore, the value of n is (1 + √(31/3)) / -2.
A parabola with equation ax^2+bx+c=y has a vertical line of symmetry at X=2 and goes through the two points (1,1) and (4,-7). The quadratic ax^2+bx+c has two real roots. The greater root is . What is n?
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