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# Two intersecting parabolas

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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)

Mar 19, 2021

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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

Mar 19, 2021