Question

find the point of intersection for the functions f(x)=-3x^2+2 and g(x)=2x+4

Answer 1

To find the point of intersection for the functions f(x)=-3x^2+2 and g(x)=2x+4, we need to set the two equations equal to each other and solve for x.

-3x^2 + 2 = 2x + 4

Rearranging the equation:

-3x^2 - 2x + 2 = 0

To solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -3, b = -2, and c = 2.

Plugging these values into the quadratic formula, we have:

x = (-(-2) ± √((-2)^2 - 4(-3)(2))) / (2(-3))
x = (2 ± √(4 + 24)) / -6
x = (2 ± √28) / -6

Simplifying further:

x = (2 ± 2√7) / -6
x = (1 ± √7) / -3

This gives us two potential x-values for the point of intersection. To find the y-values, we can substitute these x-values into either f(x) or g(x).

For f(x), using x = (1 + √7) / -3:

f((1 + √7) / -3) = -3((1 + √7) / -3)^2 + 2
= -3(1 + 2√7 + 7) / 9 + 2
= -(1 + 2√7 + 7) / 3 + 2
= -(8 + 2√7) / 3 + 2
= -8/3 - 2√7/3 + 6/3
= (-2 - 2√7 + 6) / 3
= (4 - 2√7) / 3

Therefore, one point of intersection is (x, y) = ((1 + √7) / -3, (4 - 2√7) / 3).

For g(x), using x = (1 - √7) / -3:

g((1 - √7) / -3) = 2((1 - √7) / -3) + 4
= (2 - 2√7) / -3 + 4
= -(2 - 2√7) / 3 + 12/3
= (12 - 2 + 2√7) / 3
= (10 + 2√7) / 3

Therefore, another point of intersection is (x, y) = ((1 - √7) / -3, (10 + 2√7) / 3).

So, the two points of intersection for f(x) = -3x^2 + 2 and g(x) = 2x + 4 are ((1 + √7) / -3, (4 - 2√7) / 3) and ((1 - √7) / -3, (10 + 2√7) / 3).