There are two intersecting parabolas. The first parabola has x-intercepts at X = 3 and X = 5 and passing through the point (2,-1). The second parabola has x-intercepts at X = 1 and X = 4 and, y-intercept at y = 2. Determine the EXACT x-coordinates of the points of intersection of the two parabolas. Sketch both parabolas on a graph clearly showing x-intercepts, vertex and points of intersection. (Use substitution method and quadratic formula)
+128631 The first parabola has the form
y = a ( x - 3) ( x - 5)
Since (2, -1) is on the parabola we can solve for a thusly
-1 = a ( 2 - 3) ( 2 - 5)
-1 = a ( -1)(-3)
-1 = 3a
a = -1/3
Then the parabola can be written as
y = -(1/3)(x - 3) ( x - 5)
y= -(1/3) (x^2 - 8x + 15) (1)
The second parabola has the form
y = a (x - 1)(x - 4)
Since the y intercept = 2....then the pont (0, 2) is on this graph....so
2 = a ( 0 - 1)( 0 -4)
2 = 4a
a = 2/4 = 1/2
So this parabola can be written as
y = (1/2) (x - 1) ( x - 4)
y= (1/2) ( x^2 - 5x + 4) (2)
Set (1) = (2)
(-1/3) ( x^2 -8x + 15) = (1/2)(x^2 - 5x + 4) mutiply through by 6
-2 ( x^2 - 8x + 15) = 3(x^2 -5x + 4) simplify
-2x^2 + 16x - 30 = 3x^2 - 15x + 12 rearrange as
5x^2 -31x + 42 = 0 factor as
(5x - 21) (x - 2) = 0
Set each factor to 0 and solve for x
5x - 21 = 0 x - 2 = 0
5x= 21
x= 21/5 = 4.2 x = 2
When x = 4.2 ....y = (1/2) ( 4.2^2 -5(4.2) + 4) = .32
When x =2 we know that y = -1
Here's a graph : https://www.desmos.com/calculator/9js5dz8aue
