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
, {−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
, {−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,−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}

Answer 1

To solve the equation \( x^2 + 4x - 1 = 2x + 2 \) by finding the intersection points of two functions, we can define the following functions:

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

Next, we set these functions equal to each other to find the intersection points:

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

Now, let's rearrange the equation to combine like terms:

\[ x^2 + 4x - 1 - 2x - 2 = 0 \]

This simplifies to:

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

Now, we can factor this quadratic equation:

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

Setting each factor to zero gives us the solutions:

\[ x + 3 = 0 \implies x = -3 \] \[ x - 1 = 0 \implies x = 1 \]

Thus, the solution set for the equation is:

\[ {-3, 1} \]

Therefore, the two functions used to solve the equation are:

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

And the solution set is \( { -3, 1 } \).

Correct response: f(x)=x^2+4x−1, g(x)=2x+2, {−3,1}