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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

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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