Answer:
The solutions (6, -2) and (2,2) satisfy both equations f(x) and g(x).
The solutions (2, 2) and (6, -2) are the two possible solutions of the system of equations. Geometrically, they represent the points of intersection between the parabola f(x) and the straight line g(x) on the x-y plane.
Explanation:
To solve the system of equations, we need to find the values of x that satisfy both equations f(x) and g(x).
Substitute g(x) into f(x) to eliminate one variable:
f(x) = -x^2 + 2x + 4
g(x) = -x + 4f(g(x)) = -(g(x))^2 + 2(g(x)) + 4
= -(x-4)^2 + 2(x-4) + 4
= -x^2 + 8x - 12
Now we have one equation in one variable:
-x^2 + 8x - 12 = 0
Solve for x using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = -1, b = 8, and c = -12x = (-8 ± sqrt(8^2 - 4(-1)(-12))) / 2(-1)
= (-8 ± sqrt(64 - 48)) / (-2)
= (-8 ± 4) / (-2)
= 2, 6
Therefore, the solutions of the system of equations are x = 2 and x = 6.To interpret these solutions, we can substitute them into the original equations f(x) and g(x) to find the corresponding values of y:
f(2) = -(2)^2 + 2(2) + 4 = 2
g(2) = -(2) + 4 = 2
Therefore, the solution (2, 2) satisfies both equations f(x) and g(x).
f(6) = -(6)^2 + 2(6) + 4 = -16
g(6) = -(6) + 4 = -2
Therefore, the solution (6, -2) satisfies both equations f(x) and g(x).
So, the solutions (2, 2) and (6, -2) are the two possible solutions of the system of equations. Geometrically, they represent the points of intersection between the parabola f(x) and the straight line g(x) on the x-y plane.