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.
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) .