Which is the intersection of the parabola y=2x^2 and the line y=8x-6?
To find the intesection with the function and the line we need to obtain the point(s) in wich they intersect:
in the fuction y = 2x^2 , substitute y value with 8x-6
then , 8x-6 = 2x^2
==> 2x^2 -8x +6 = 0
==> (2x-6)(x-1) = 0
then x values are :
x1 = 3 and x2 = 1
now y values are :
y = 8x -6
then, y1= 18 and y2 = 2
then the function and the line intersect at the points :
( 3, 18) and (1, 2)
It is known that the intersection of the graphs consists of the common points of the graphs, these common points being found by solving the equations of the graphs, simultaneously.
In our case, we'll substitute in the equation of the parabola, the unknown y, by the expression from the equation of the line.
8x-6 = 2x^2
We'll move the terms from the left side, to the right side, with opposite value:
2x^2 - 8x + 6 = 0
We'll use the formula for the quadratic equations to find out the roots.
x1 = [(8+sqrt(64-48)]/4
x1 = (8+sqrt16)/4
x1 = (8+4)/4
x1 = 12/4
x1 = 3, so y1 = 8x1 - 6 = 8*3-6 = 24-6 = 18
x2 = (8-4)/4
x2 = 1, so y2 = 8x2 - 6 = 8*1 - 6 = 8-6 = 2
The common point are:
(3,18) and (1,2)
y = 2x^2 (1) and y = 8x-6. (2)
Therefore eliminating y between the 2 equations, we get:
8x-6 = 2x^2 . Or dividing by2,
4x-3 =x^2 . Or x^2-4x+3 . Or (x-3)(x-1) = 0. So x =1 and x =4.
When x =1, y = 2 and when x =3. y = 2*3^2 = 18.
So the points of intersections are (1.2) and (3,18)