Solve fx gx What are the points of intersection of the grap

Solve f(x) = g(x). What are the points of intersection of the graphs of the two functions? f(x) = -3x^2 + 6 g(x) = 4x + 7 If f(x) = g(x), then x =. (Simplify your answer. Use a comma to separate answers as needed.) The point(s) of intersection of the two graphs is/are. (Simply your answer. Type an ordered pair. Use a comma to separate answers as needed.)

Solution

Equating \"

-3x^2 + 6 = 4x + 7

3x^2 + 4x + 1 = 0

Now, this is a quadratic of the form ax^2 + bx + c
which can be factored(sometimes, but here it can be factored).

a = 3, b= 4 and c = 1

Now, we need two numbers that sum to b
Sum = 4

And those numbers multiply to ac
Multiply to 3(1) = 3

Sum = 4 and product = 3

This happens with the numbers 3 and 1...

So, we have :

3x^2 + 3x + x + 1

3x(x + 1) + 1(x + 1) = 0

(3x + 1)(x + 1) =0

3x + 1 = 0 and x + 1= 0

x = -1/3 and x =-1

x = -1 , -1/3

Now, when x = -1 y = -4 + 7 = 3
When x = -1/3, y = -4/3 + 7 = 17/3

So, answers :

x = -1 , -1/3
(-1 , 3) , (-1/3 , 17/3)