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

x2+4x−1=2x+2

(1 point)
Responses

f(x)=x2+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)=x2+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)=x2+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

f(x)=x2+4x−1
, g(x)=2x+2
, {−1,3}

Answer 1

To solve the equation \(x^2 + 4x - 1 = 2x + 2\), we can rewrite it into a standard function form:

1. Rearrange the equation to one side: \[ x^2 + 4x - 1 - 2x - 2 = 0 \] Which simplifies to: \[ x^2 + 2x - 3 = 0 \]

2. We can create two functions based on the original equation: \[ f(x) = x^2 + 4x - 1 \] \[ g(x) = 2x + 2 \]

3. To find the intersection points of these two functions, we will find where \(f(x) = g(x)\), which leads us back to the equation we derived: \[ x^2 + 4x - 1 = 2x + 2 \] This reduces to: \[ x^2 + 2x - 3 = 0 \]

4. Next, we can factor or use the quadratic formula to find the roots: Factoring gives us: \[ (x + 3)(x - 1) = 0 \] This gives us the solutions: \[ x = -3 \quad \text{and} \quad x = 1 \]

5. Therefore, the solutions of the original equation, or the intersection points of the functions \(f(x)\) and \(g(x)\), are: \[ {-3, 1} \]

Thus, your final response would be: f(x)=x^2+4x−1, g(x)=2x+2, {−3,1}