Two parabolas are the graphs of the equations y=2x^2-7x+1 and y=8x^2+6x+1. Give all points where they intersect. List the points in order of increasing x-coordinate, separated by semicolons.
+128408 Set the equations equal
8x^2 + 6x + 1 = 2x^2 - 7x + 1 rearrange as
6x^2 + 13x = 0 factor
x (6x - 13) = 0
The x coordinates are
x = 0 and 6x - 13 = 0
6x = 13
x =13/6
When x = 0 y =1
When x = 13/6, y= 2(13/6)^2 - 7(13/6) + 1 = -43/9
The points of intersection are
(0 , 1) , (13/6 , -43/9 )
+2666 Set equal: \(2x^2-7x+1 =8x^2+6x+1\)
Simplify: \(6x^2 + 13x = 0\)
Factor: \(x(6x + 13) = 0\)
If \(6x + 13 = 0\), \(x = -{13 \over 6}\), else \(x = 0\)
Substituting this into the quadratic, we find the points of intersection are \(\color{brown}\boxed{(-{13 \over 6},{ 43 \over 9}) \ \text{and} \ ({0, 1})}\)
