+0 The line y=3 intersects the graph of y = 4x2+x-1 at the points A and B. The distance between A and B can be written as \(\sqrt{m}/n\), where m and n are positive integers that do not share any factors other than one. Find the value of m-n.
+129830 Set the equations equal
3 = 4x^2 + x - 1 simplify
4x^2 + x - 4 = 0 divide through by 4
x^2 - x/4 - 1 = 0
x^2 + x/4 = 1 complete the square on x
x^2 + x/4 + 1/64 = 1 + 1/64
(x + 1/8)^2 = 65/64 take both roots
x + 1/8 = sqrt(65) / 8 x + 1/8 = -sqrt (65) / 8
x = [ sqrt (65) - 1 ] / 8 x = [-sqrt (65) - 1 ] / 8
A = ( [-sqrt (65) - 1 ] / 8 , 3 )
B = ([ sqrt (65) - 1 ] / 8 , 3 )
The distance between these points = 2sqrt (65) / 8 = sqrt (65) / 4
m - n = 65 - 4 = 61
