To show how to solve the equation, create two functions and find their intersection points. What two functions can be used to solve the following equation, and what is their solution set?

Question

x^2+4x−1=2x+2

(1 point)
Responses

f(x)=x^2+4x−1
, g(x)=2x+2
, {−1,3}
f left parenthesis x right parenthesis equals x squared plus 4 x minus 1 , g left parenthesis x right parenthesis equals 2 x plus 2 , left brace negative 1 comma 3 right brace

f(x)=x^2+4x−1
, g(x)=2x+2
, {−3,1}
f left parenthesis x right parenthesis equals x squared plus 4 x minus 1 , g left parenthesis x right parenthesis equals 2 x plus 2 , left brace negative 3 comma 1 right brace

f(x)=x^2+4x−1
, g(x)=2x+2
, {−3,−4,1,4}
f left parenthesis x right parenthesis equals x squared plus 4 x minus 1 , g left parenthesis x right parenthesis equals 2 x plus 2 , left brace negative 3 comma negative 4 comma 1 comma 4 right brace

f(x)=x^2+4x
, g(x)=2x
, {−3,1}
f left parenthesis x right parenthesis equals x squared plus 4 x , g left parenthesis x right parenthesis equals 2 x , left brace negative 3 comma 1 right brace

Answer 1

To solve the equation \( x^2 + 4x - 1 = 2x + 2 \), we can rearrange it into the form suitable for finding intersection points of two functions.

1. Rearranging the equation:

   \[ x^2 + 4x - 1 - 2x - 2 = 0 \] Simplifying this gives: \[ x^2 + 2x - 3 = 0 \]

2. Now, we can define two functions:

   \[ f(x) = x^2 + 4x - 1 \] \[ g(x) = 2x + 2 \]

   Alternatively, you could define \( f(x) \) as \( x^2 + 2x - 3 \) and set it equal to zero, but in this case, we are instructed to use the original forms of the equations.

3. To find the intersection of these two functions, we set \( f(x) = g(x) \):

   \[ x^2 + 4x - 1 = 2x + 2 \]

   Rearranging gives us the quadratic equation:

   \[ x^2 + 2x - 3 = 0 \]

4. Now we can solve \( x^2 + 2x - 3 = 0 \) using factoring:

   \[ (x + 3)(x - 1) = 0 \]

   Setting each factor to zero gives us the solutions:

   \[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \] \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]

So, the solution set is: \[ {-3, 1} \]

Thus, the response that corresponds is: f(x)=x^2+4x−1, g(x)=2x+2, {−3,1}