Solve f(x)g(x). what are the points of intersection of the graphs of the two functions?
f(x) = x^2 -x + 2 ; g(x) = 2x^2 - 3x - 22
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The points of intersection of the graphs of two functions f(x) and g(x) are the solutions to the equation f(x) = g(x).
So, we set f(x) = g(x) and solve for x:
x^2 - x + 2 = 2x^2 - 3x - 22
Rearranging terms, we get:
2x^2 - x^2 - 3x + x - 22 - 2 = 0
Simplifying, we get:
x^2 - 2x - 24 = 0
This is a quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = -2, and c = -24. We can solve this equation using the quadratic formula x = [-b ± sqrt(b^2 - 4ac)] / (2a):
x = [2 ± sqrt((-2)^2 - 4*1*(-24))] / (2*1)
x = [2 ± sqrt(4 + 96)] / 2
x = [2 ± sqrt(100)] / 2
x = [2 ± 10] / 2
So the solutions are x = 12/2 = 6 and x = -8/2 = -4.
Therefore, the points of intersection of the graphs of the two functions are (6, f(6)) and (-4, f(-4)).
To find the y-coordinates of these points, we substitute x = 6 and x = -4 into the equation for f(x):
f(6) = (6)^2 - 6 + 2 = 36 - 6 + 2 = 32
f(-4) = (-4)^2 - (-4) + 2 = 16 + 4 + 2 = 22
So the points of intersection are (6, 32) and (-4, 22).
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