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Two parabolas are the graphs of the equations $y=3x^2+4x-5$ and $y=x^2+11$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons.

 Jun 17, 2018
 #1
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A point of intersection is a point that makes both equations true.

 

y  =  3x2 + 4x - 5       and       y  =  x2 + 11

 

Starting with the first equation...
 

y  =  3x2 + 4x - 5

                                       Since  y  =  x2 + 11  we can substitute  x2 + 11  in for  y .

x2 + 11  =  3x2 + 4x - 5

                                       Subtract  x2  from both sides of the equation.

11  =  2x2 + 4x - 5

                                       Subtract  11  from both sides of the equation.

0  =  2x2 + 4x - 16

                                       Divide through by  2 .

0  =  x2 + 2x - 8

                                       Factor the right side.

0  =  (x - 2)(x + 4)

                                       Set each factor equal to zero and solve for  x .

 

x - 2  =  0          or          x + 4  =  0

 x  =  2              or             x  =  -4

 

Use these values of  x  to find  y .

 

If    x  =  2    then    y  =  (2)2 + 11  =  4 + 11  =  15    So   (2, 15)  is a point of intersection.

 

If    x  =  -4    then    y  =  (-4)2 + 11  =  16 + 11  =  27    So   (-4, 27)  is a point of intersection.

 

The points of intersection are   (-4, 27)   and   (2, 15) .

 Jun 18, 2018

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